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Kinetic models in natural sciences Jacek Banasiak1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa [email protected] 1 Introduction 1.1 Preliminaries In our terminology, a kinetic type equation describes an evolution of a population of objects depending on certain attributes from a certain space, subject to a given set of conservation laws. One of the natural ways to describe such a population is by providing the density of the objects with respect to the attributes and investigate how it changes in time. The density, say u(x), is either the number of elements with an attribute x (if the number of possible attributes is finite or countable), or a gives the number of elements with attributes in a set A, according to the formula ∫ u(x)dµ, (1) A if x is a continuous variable. In many cases we are interested in tracking the total number of elements of the population which, for a time t, is given by ∑ u(x, t), (2) x∈Ω if Ω is countable, and ∫ u(x, t)dx, (3) Ω if Ω is a continuum, where the set Ω is the space of attributes. Kinetic equations for u usually are built in the following way. Let u(x, t) be the density of a quantity Q with respect to the attribute(s) x from Ω at a time t. Then the equation is obtained by balancing, for any subset A of the space of attributes, 2 Jacek Banasiak 1. the loss of Q from A due to the change of their attributes to due internal interactions; 2. the gain of Q in A due to the changes of individuals’ attributes outside A to the ones from A; 3. transport through A. This results in the ‘Master Equation’ ∂t u(x, t) = [Ku](x, t) := [T u](x, t) + [Au](x, t) + [Bu](x, t), (4) where A is the loss operator, B is the gain operator, while the T describes transport phenomena. Equation (4) is supplemented with the initial state of the system ◦ u(x, 0) = u (x), x ∈ Ω. (5) Only in exceptional circumstances can the problem (4), (5) be solved. Usually, we have to prove the existence, uniqueness and other relevant properties of the solution u without knowing its explicit form. There are various ways of doing this. We shall follow the dynamical systems approach. Here, the evolution of the system using a family of operators (G(t))t≥0 , parameterised by time, that ◦ map an initial state u of the system to all subsequent states in the evolution; that is, the solutions are represented as ◦ u(t) = G(t) u . (6) The solutions, or the states of the system, belong to some appropriate state space which is chosen partly due to its relevance to the problem but also for the mathematical convenience. By no means is this choice unique: it is a mathematical intervention into the model. In the processes discussed in these lectures, an appropriately defined integral of the density over the space of attributes is the the total amount of individuals, or the total mass of the system. Due to the conservation laws used to construct the equation, this integral is constant, or changes in some pre-defined way. From this point of view it is natural to consider such processes as evolutions of densities in the so-called L1 spaces; that is, in ∫ L1 (Ω, µ) = u; ∥u∥ = |u|dµ < +∞ , Ω where µ is an appropriate measure corresponding to the process. Such a space will be our state space; that is, the state of the system will be described by a density with finite total mass. However, we can try to control the process using some other gauge function. For instance, if we were interested in controlling the maximal concentration of the particles, a more proper choice would be to use the functional Kinetic models in natural sciences 3 sup |u(x)| x∈Ω as the gauge function. This approach leads in a natural way to a class of abstract spaces called the Banach spaces. Interlude – Banach spaces and linear operators In what follows we shall restrict our attention to the state spaces which are Banach spaces. To recall, a Banach space is a vector space X, equipped with a finite gauge function ∥ · ∥, called norm, satisfying ∥x∥ = 0 if and only if x = 0, ∥αx∥ = |α|∥x∥ for each scalar α and ∥x + y∥ ≤ ∥x∥ + ∥y∥, x, y ∈ X and which is complete with respect to the convergence defined by the norm (a space is complete if it contains limits of all Cauchy sequences). Example 1. We introduce another type of Banach spaces which will be used throughout the lectures: the Sobolev spaces. They are constructed on the basis of L1 (Ω). In general considerations, when dealing with partial derivatives of functions, often only the order of the derivative is important. In such cases, to shorten calculations, we introduce the following notation. Let α = (α1 , . . . , αn ), αi ∈ N0 , i = 1, . . . , n, be a multi-index and denote |α| = α1 + · · · + αn . Then, for a given (locally integrable) function u we denote any generalized (distributional) derivative of u of order |α| by ∂αu = ∂ |α| αn u. 1 ∂xα 1 · · · ∂xn The Sobolev spaces W1m (Ω) are defined as W1m (Ω) := {u ∈ L1 (Ω); ∂ α u ∈ L1 (Ω), |α| ≤ m} . In the same way, starting from the space Lp (Ω) of functions integrable with power p, we can define Sobolev spaces Wpm (Ω), p ∈ [1, ∞[. For p = ∞ the corresponding space L∞ (Ω) is the space of functions which are bounded almost m (Ω) is the space in which all generalized derivatives everywhere on Ω, and W∞ up to the order m have this property as well. An object intimately related with a Banach space is a linear operator. A (linear) operator from X to Y is a linear function A : D(A) → Y , where D(A) is a linear subspace of X, called the domain of A. We use the notation (A, D(A)) to denote the operator A with domain D(A). By L(X, Y ) we denote the space of all bounded operators between X and Y ; that is, the operators for which ∥A∥ := sup ∥Ax∥ = sup ∥Ax∥ < +∞. (7) ∥x∥≤1 ∥x∥=1 4 Jacek Banasiak The space L(X, X) is abbreviated as L(X). We further define the kernel Ker A (or the null-space of A) by Ker A = {x ∈ D(A); Ax = 0} and the range Ran A of A by Ran A = {y ∈ Y ; Ax = y for some x ∈ D(A)} Furthermore, the graph of A is defined as the set {(x, y) ∈ X × Y ; x ∈ D(A), y = Ax}. We say that the operator A is closed if its graph is a closed subspace of X × Y . Equivalently, A is closed if and only if for any sequence (xn )n∈N ⊂ D(A), if lim xn = x in X and lim Axn = y in Y , then x ∈ D(A) n→∞ n→∞ and y = Ax. An operator A in X is closable if the closure of its graph is itself a graph of an operator. In such a case the operator whose graph is G(A) is called the closure of A and denoted by A. Example 2. Consider the operator Af = f ′ in C([0, 1]) and L1 ([0, 1]). Then, [6, Example 2.3], A is unbounded in both spaces, closed in C([0, 1]) and not closed, but closable, in L1 ([0, 1]). In this way, (4) can be written as the Cauchy problem for an ordinary differential equation in an appropriate Banach space X: find R+ ∋ t → u(t) ∈ X such that ◦ ∂t u = Ku, t > 0, u(0) = u∈ X, (8) where K : D(K) → X is a realization of the expression K, defined on some subset D(K) of the chosen state space X. It is clear that a minimum requirement for D(K) is that [Ku](·) ∈ X for u ∈ D(K). It is important to remember that the expression K usually has multiple realizations and finding an appropriate one, such that with (K, D(K)) the problem (8) is well posed (often called the generator of the process) is a very difficult task. We mention the so-called maximal realization of the expression K, Kmax defined as the restriction of K to D(Kmax ) = {u ∈ X; x → [Ku](x) ∈ X}. The generator may be, or may be not, equal to Kmax . In the former case, typically (4) is uniquely solvable in X. 1.2 The models In this section we shall discuss the examples which will be discussed in the course. Kinetic models in natural sciences 5 1.3 Transport on networks The first example does not exactly fit into (4). We consider a network with some substance flowing along the edges and being redistributed in the nodes. The process of redistribution of the flow is the loss-gain process governed by the Kirchoff’s law (flow-in = flow-out) and thus is an example of a kinetic process as defined above. The network under consideration is represented by a simple directed graph G = (V (G), E(G)) = ({v1 , . . . , vn }, {e1 , . . . , em }) with n vertices v1 , . . . , vn and m edges (arcs), e1 , . . . , em . We suppose that G is connected but not necessarily strongly connected. Each edge is normalized so as to be identified with [0, 1] with the head at 0 and the tail at 1. The outgoing incidence matrix, Φ− = (ϕ− ij )1≤i≤n,1≤j≤m , and the incoming incidence matrix, + + Φ = (ϕij )1≤i≤n,1≤j≤m , of this graph are defined, respectively as ϕ− ij = { { ej 1 if vi → 0 otherwise. ϕ+ ij = ej 1 if → vi 0 otherwise. If the vertex vi has more than one outgoing edge, we place a non negative weight wij on the outgoing edge ej such that for this vertex vi , ∑ wij = 1, j∈Ei where Ei is defined by saying that j ∈ Ei if the edge ej is outgoing from vi . Naturally, wij = 1 if Ei = {j} and, to shorten notation, we adopt the convention that wij = 1 for any j if Ei = ∅. Then the weighted outgoing − by replacing each nonzero ϕ− incidence matrix, Φ− w , is obtained from Φ ij entry by wij . If each vertex has an outgoing edge, then Φ− w is row stochastic, T hence Φ− (Φ− w ) = In (where the superscript T denotes the transpose). The (weighted) adjacency matrix A = (aij )1≤i,j≤n of the graph is defined be taking ek aij = wjk if there is ek such that vj → vi and 0 otherwise; that is, A = + − T Φ (Φw ) . An important role is played by the line graph Q of G. To recall Q = (V (Q), E(Q)) = (E(G), E(Q)), where E(Q) = {uv; u, v ∈ E(G), the head of u coincides with the tail of v} = {εj }1≤j≤k . By B we denote the weighted adjacency matrix for the line graph; that is, T + B = (Φ− w) Φ . (9) If there is an outgoing edge at each vertex then, from the definition of B, we see that it is column stochastic. A vertex v will be called a source if there are no incoming edges towards it and a sink if there are no edges outgoing from it. 6 Jacek Banasiak Example 1.1. Consider the following graph. For Φ− , Φ+ are given below. 00 1100000 1 0 0 0 1 1 0 0 0 + Φ− = 0 0 0 0 1 0 0, Φ = 0 1 0 0 0 0 0 1 0 0 0 00 0000001 while the adjacency matrix is given by 0000 1 0 0 0 1 1 0 1 0 0 0 0 0110 this graph, the matrices 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 , 1 0 0 0 0 . 1 0 We are interested in a flow on a closed network G. Then the standard assumption is that the flow satisfies the Kirchoff law at the vertices m ∑ j=1 ϕ− ij cj uj (1, t) = wij m ∑ ϕ+ ij cj uj (0, t), t > 0, i ∈ 1, . . . , n, j=1 which, in this context, is the conservation of mass law: the total inflow of mass per unit time equals the total outflow at each node (vertex) of the network. Let uj (x, t) be the density of particles at position x and at time t ≥ 0 flowing along edge ej for x ∈ [0, 1]. The particles on ej are assumed to move with velocity cj > 0 which is constant for each j. We consider a generalized Kirchoff’s law by allowing for decrease/amplification of the flow at the entrances and exits at each vertex. Then the flow in the system is described by ∂ u (x, t) = cj ∂x uj (x, t), x ∈ (0, 1), t ≥ 0, t j uj (x, 0) = fj (x), (10) m ∑ − ϕ+ ϕij ξj cj uj (1, t) = wij ik (γk ck uk (0, t)), k=1 where γj > 0 and ξj > 0 are the absorption/amplification coefficients at, respectively, the head and the tail of the edge ej . If γj = ξj = 1 for all j = 1, · · · , m, then the boundary conditions simply describe the Kirchoff law at the vertices. Kinetic models in natural sciences 7 Remark 1. We observe that the boundary condition in (10) takes a special form if vi is either a sink or a source. If it is a sink, then Ei = ∅ and, by the convention above, 0= m ∑ ϕ+ ik (γk ck uk (0, t)), t > 0, (11) k=1 and ϕ− ij ξj cj uj (1, t) = 0, t > 0, j = 1, . . . , m, (12) if it is a source. Clearly the last condition is nontrivial only if j ∈ Ei as then ϕ− ij ̸= 0. Let us denote C = diag(cj )1≤j≤m , K = diag(ξj )1≤j≤m and G = diag(γj )1≤j≤m . We consider (10) as an abstract Cauchy problem ut = Au, u(0) = f , (13) in X = (L1 ([0, 1]))m , where A is the realization of the expression A = (cj ∂x )1≤j≤m on the domain D(A) = {u ∈ (W11 ([0, 1]))m ; u satisfies the boundary conditions in (10)}. (14) It can be proved that D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BGCu(0)}. (15) 1.4 Epidemiological models with age structure The gain and loss part of the model are given by the SIRS system S ′ = −Λ(I)S + δI, I ′ = Λ(I)S − (δ + γ)I, R′ = γI, (16) where S, I, R are, respectively, the number of susceptibles, infectives and recovered (with immunity) and γ, δ are recovery rates with and without immunity. For many diseases the rates of infection and recovery significantly vary with age. Thus the vital dynamics of the population and the infection mechanism interact to produce a nontrivial dynamics. To model it, we assume that the total population in the absence of disease can be modelled by the linear McKendrick model describing the evolution in time of the density of the population with respect to age a ∈ [0, ω], ω < ∞, denoted by n(a, t). The evolution is driven by the processes of death and birth with vital rates µ(a) and β(a), respectively. Due to epidemics, we split the population into susceptibles, infectives and recovered, 8 Jacek Banasiak n(a, t) = s(a, t) + i(a, t) + r(a, t), so that the scalar McKendrick equation for n splits, according to (16), into the system ∂t s(a, t) + ∂a s(a, t) + µ(a)s(a, t) = −Λ(a, i(·, t))s(a, t) + δ(a)i(a, t), ∂t i(a, t) + ∂a i(a, t) + µ(a)i(a, t) = Λ(a, i(·, t))s(a, t) − (δ(a) + γ(a))i(a, t), ∂t r(a, t) + ∂a r(a, t) + µ(a)r(a, t) = γ(a)i(a, t), (17) where now the rates are age specific, see [21]. The function Λ is the infection rate (or the force of infection). In the the intercohort model we use ∫ω Λ(a, i(·, t)) = K(a, a′ )i(a′ , t)da′ , (18) 0 where K is a nonnegative bounded function which accounts for the age dependence of the infections. For instance, for a typical childhood disease, K should be large for small a, a′ and close to zero for large a or a′ (not necessarily 0, as usually adults can contract them). The extreme case, of an intracohort infection, is given by Λ(a, i(·, t)) = K0 (a)i(a, t). (19) System (17) is supplemented by the boundary conditions ∫ω β(a)(s(a, t) + (1 − p)i(a, t) + (1 − q)r(a, t))da, s(0, t) = 0 ∫ω i(0, t) = p β(a)i(a, t)da, 0 ∫ω r(0, t) = q β(a)r(a, t)da, (20) 0 where p, q ∈ [0, 1] are the vertical transmission parameters of infectiveness and immunity, respectively. Finally, we prescribe the initial conditions ◦ s(a, 0) =s (a), ◦ i(a, 0) = i (a), ◦ r(a, 0) =r (a). (21) Fragmentation-coagulation processes They may seem very specific, but the range of applications is vast: • Chemical engineering: polymerization/depolimerization processes, with possible mass loss through dissolution, chemical reactions, oxidation etc, or mass growth due to the deposition of material on the clusters. Kinetic models in natural sciences • • • 9 Biology: Blood cells’ coagulation and splitting, animal grouping, phytoplankton at the level of aggregates. Planetology: merging of planetesimals. Aerosol research: coagulation of smoke, smog and dust particles, droplets in clouds. Also, they are most rewarding kinetic processes to study from analytical point of view. In the case of pure fragmentation standard modelling process leads to the following equation: ∫∞ ∂t u(x, t) = −a(x)u(x, t) + a(y)b(x|y)u(y, t)dy, (22) x u is the density of particles of mass x, a is the fragmentation rate and b describes the distribution of particle masses x spawned by the fragmentation of a particle of mass y. Further ∫∞ M (t) = xu(x, t)dx (23) 0 is the total mass of the ensemble at time t. Local conservation principle requires ∫y xb(x|y)dx = y, (24) 0 with the expected number of particles produced by a particle of mass y is given by ∫y n0 (y) = b(x|y)dx. 0 Fragmentation can be supplemented by growth/decay, transport or diffusion processes, [6, 12, 13], but we will not discuss them here. If we combine the fragmentation process with coagulation, we will get ∫∞ ∂t u(x, t) = −a(x)u(x, t) + a(y)b(x|y)u(y, t)dy (25) x ∫∞ −u(x, t) k(x, y)u(y, t)dy + 0 1 2 ∫x k(x − y, y)u(x − y, t)u(y, t)dy. 0 The coagulation kernel k(x, y) represents the likelihood of a particle of size x attaching itself to a particle of size y and, for a moment, we assume that it is a symmetric nonnegative positive function. 10 Jacek Banasiak Since the fragmentation and coagulation processes just rearrange the mass distribution among the clusters, (23) implies that the natural space for to analyse the fragmentation-coagulation processes is ∫∞ X1 = L1 (R+ , xdx) = u; ∥u∥1 = |u(x)|xdx < +∞ . 0 However, for technical reasons, for coagulation processes it is important to control also the number of particles, or even higher moments of the density. The best results are obtained in spaces ∫∞ X1,α = L1 (R+ , (1 + xα )dx) = u; ∥u∥0,α = |u|(1 + xα )dx < +∞ , 0 α ≥ 1. 2 The tools 2.1 Basic positivity concepts The common feature of the introduced models is that the solution originating from a nonnegative density should stay nonnegative; that is, the solution operator should be a ‘positive’ operator. Since we are talking about general Banach spaces, we have to define what we mean by a nonnegative element of a Banach space. Though in all cases discussed here our Banach space is an L1 (Ω, µ) space, where the nonnegativity of a function f is understood as f (x) ≥ 0 µ-almost everywhere, it is more convenient to work in a more abstract setting. Defining Order In a given vector space X an order can be introduced either geometrically, by defining the so-called positive cone (in other words, what it means to be a positive element of X), or through the axiomatic definition: Definition 1. Let X be an arbitrary set. A partial order (or simply, an order) on X is a binary relation, denoted here by ‘ ≥’, which is reflexive, transitive, and antisymmetric, that is, (1) x ≥ x for each x ∈ X; (2) x ≥ y and y ≥ x imply x = y for any x, y ∈ X; (3) x ≥ y and y ≥ z imply x ≥ z for any x, y, z ∈ X. Kinetic models in natural sciences 11 The supremum of a set is its least upper bound and the infimum is the greatest lower bound. The supremum and infimum of a set need not exist. For a two-point set {x, y} we write x ∧ y or inf{x, y} to denote its infimum and x ∨ y or sup{x, y} to denote supremum. We say that X is a lattice if every pair of elements (and so every finite collection of them) has both supremum and infimum. From now on, unless stated otherwise, any vector space X is real. Definition 2. An ordered vector space is a vector space X equipped with partial order which is compatible with its vector structure in the sense that (4) x ≥ y implies x + z ≥ y + z for all x, y, z ∈ X; (5) x ≥ y implies αx ≥ αy for any x, y ∈ X and α ≥ 0. The set X+ = {x ∈ X; x ≥ 0} is referred to as the positive cone of X. If the ordered vector space X is also a lattice, then it is called a vector lattice or a Riesz space. For an element x in a Riesz space X we can define its positive and negative part, and its absolute value, respectively, by x+ = sup{x, 0}, x− = sup{−x, 0}, |x| = sup{x, −x}, which are called lattice operations. We have x = x+ − x− , |x| = x+ + x− . (26) The absolute value has a number of useful properties that are reminiscent of the properties of the scalar absolute value. As the next step, we investigate the relation between the lattice structure and the norm, when X is both a normed and an ordered vector space. Definition 3. A norm on a vector lattice X is called a lattice norm if |x| ≤ |y| implies ∥x∥ ≤ ∥y∥. (27) A Riesz space X complete under the lattice norm is called a Banach lattice. Property (27) gives the important identity: ∥x∥ = ∥|x|∥, x ∈ X. (28) AM - and AL-spaces Two important classes of Banach lattices that play a significant role later are provided by the AL- and AM - spaces. Definition 4. We say that a Banach lattice X is (i) an AL-space if ∥x + y∥ = ∥x∥ + ∥y∥ for all x, y ∈ X+ , 12 Jacek Banasiak (ii) an AM-space if ∥x ∨ y∥ = max{∥x∥, ∥y∥} for all x, y ∈ X+ . Example 3. Standard examples of AM -spaces are offered by the spaces C(Ω), where Ω is either a bounded subset of Rn , or in general, a compact topological space. Also the space L∞ (Ω) is an AM -space. On the other hand, most known examples of AL-spaces are the spaces L1 (Ω). These examples exhaust all (up to a lattice isometry) cases of AM - and AL-spaces. However, particular representations of these spaces can be very different. 2.2 Positive Operators Definition 5. A linear operator A from a Banach lattice X into a Banach lattice Y is called positive, denoted by A ≥ 0, if Ax ≥ 0 for any x ≥ 0. An operator A is positive if and only if |Ax| ≤ A|x|. This follows easily from −|x| ≤ x ≤ |x| so, if A is positive, then −A|x| ≤ Ax ≤ A|x|. Conversely, taking x ≥ 0, we obtain 0 ≤ |Ax| ≤ A|x| = Ax. A frequently used property of positive operators is given in the following theorem. Theorem 1. If A is an everywhere defined positive operator from a Banach lattice to a normed Riesz space, then A is bounded. Proof. If A were not bounded, then we would have a sequence (xn )n∈N satisfying∑∥xn ∥ = 1 and ∥Axn ∥ ≥ n3 , n ∈ N. Because X is a Banach space, ∞ x := n=1 n−2 |xn | ∈ X. Because 0 ≤ |xn |/n2 ≤ x, we have ∞ > ∥Ax∥ ≥ ∥A(|xn |/n2 )∥ ≥ ∥A(xn /n2 )∥ ≥ n for all n, which is a contradiction. ⊓ ⊔ The norm of a positive operator can be evaluated by ∥A∥ = sup x≥0, ∥x∥≤1 ∥Ax∥. (29) Indeed, since ∥A∥ = sup∥x∥≤1 ∥Ax∥ ≥ supx≥0,∥x∥≤1 ∥Ax∥, it is enough to prove the opposite inequality. For each x with ∥x∥ ≤ 1 we have |x| = x+ + x− ≥ 0 with ∥x∥ = ∥|x|∥ ≤ 1. On the other hand, A|x| ≥ |Ax|, hence ∥A|x|∥ ≥ ∥|Ax|∥ = ∥Ax∥. Thus sup∥x∥≤1 ∥Ax∥ ≤ supx≥0,∥x∥≤1 ∥Ax∥ and the statement is proved. As a consequence, we note that if 0 ≤ A ≤ B ⇒ ∥A∥ ≤ ∥B∥. (30) Moreover, it is worthwhile to emphasize that if there exists K such that ∥Ax∥ ≤ K∥x∥ for x ≥ 0, then this inequality holds for any x ∈ X. Indeed, by (29) we have ∥A∥ ≤ K and using the definition of the operator norm, we obtain the desired statement. Kinetic models in natural sciences 13 2.3 Relation Between Order and Norm There is a useful relation between the order, norm (absolute value) and the convergence of sequences in R – any monotonic sequence which is bounded (in absolute value), converges. One would like to have a similar result in Banach lattices. It turns out to be not so easy. Existence of an order in some set X allows us to introduce in a natural way the notion of (order) convergence. Proper definitions of order convergence require nets of elements but we do not need to go to such details. For a non-increasing sequence (xn )n∈N we write xn ↓ x if inf{xn ; n ∈ N} = x. For a non-decreasing sequence (xn )n∈N the symbol xn ↑ x have an analogous meaning. Then we say that an arbitrary sequence (xn )n∈N is order convergent to x if it can be sandwiched between two monotonic sequences o converging to x. We write this as xn → x. One of the basic results is: Proposition 1. Let X be a normed lattice. Then: (1) The positive cone X+ is closed. (2) If (xn )n∈N is nondecreasing and limn→∞ xn = x in the norm of X, then x = sup{xn ; n ∈ N}. Analogous statement holds for nonincreasing sequences. In general, the converse of Proposition 1(2) is false; that is, we may have xn ↑ x but (xn )n∈N does not converge in norm. Indeed, consider xn = (1, 1, 1 . . . , 1, 0, 0, . . .) ∈ l∞ , where 1 occupies only the n first positions. Clearly, supn∈N xn = x := (1, 1, . . . , 1, . . .) but ∥xn − x∥∞ = 1. Such a converse holds in a special class of Banach lattices, called Banach lattices with order continuous norm. There we have, in particular, that 0 ≤ xn↑ x and xn ≤ x for all n ∈ N if and only if (xn )n∈N is a Cauchy sequence, [1, Theorem 12.9]. All Banach lattices Lp (Ω) with 1 ≤ p < ∞ have order continuous norms. On the other hand, neither L∞ (Ω) nor spaces of continuous functions C(Ω̄) (if Ω does not consist of isolated points) have order continuous norm. The requirement that (xn )n∈N must be order dominated often is too restrictive. The spaces we are mostly concerned belong to a class which have a stronger property. Definition 6. We say that a Banach lattice X is a KB-space (Kantorovič– Banach space) if every increasing norm bounded sequence of elements of X+ converges in norm in X. We observe that if xn ↑ x, then ∥xn ∥ ≤ ∥x∥ for all n ∈ N and thus any KB-space has order continuous norm. Hence, spaces which do not have order continuous norm cannot be KB-spaces. This rules out the spaces of continuous functions, l∞ and L∞ (Ω) from being KB-spaces. 14 Jacek Banasiak Any reflexive Banach space is a KB-space, [6, Theorem 2.82]. That ALspaces (so, in particular, all L1 spaces) are also KB-spaces follows from the following simple argument. Theorem 2. Any AL-space is a KB-space. Proof. If (xn )n∈N is an increasing and norm bounded sequence, then for 0 ≤ xn ≤ xm , we have ∥xm ∥ = ∥xm − xn ∥ + ∥xn ∥ as xm − xn ≥ 0 so that ∥xm − xn ∥ = ∥xm ∥ − ∥xn ∥ = |∥xm ∥ − ∥xn ∥| . Because, by assumption, (∥xn ∥)n∈N is monotonic and bounded, and hence convergent, we see that (xn )n∈N is a Cauchy sequence and thus converges. ⊓ ⊔ 2.4 Complexification All our models refer to models, solutions of which must be real. Thus, our problems should be posed in real Banach spaces. However, to take full advantage of the tools of functional analysis, such as the spectral theory, it is worthwhile to extend our spaces to include also complex valued functions, so that they become complex Banach spaces. While the algebraic and metric structure of Banach spaces can be easily extended to the complex setting, the extension of the order structure must be done with more care. This is done by the procedure called complexification. Definition 7. Let X be a real vector lattice. The complexification XC of X is the set of pairs (x, y) ∈ X × X where, following the scalar convention, we write (x, y) = x + iy. Vector operations are defined as in scalar case x1 + iy1 + x2 + iy2 = x1 + x2 + i(y1 + y2 ), (α + iβ)(x + iy) = αx − βy + i(βx + αy). The partial order in XC is defined by x0 + iy0 ≤ x1 + iy1 if and only if x0 ≤ x1 and y0 = y1 . (31) Remark 2. Note, that from the definition, it follows that x ≥ 0 in XC is equivalent to x ∈ X and x ≥ 0 in X. In particular, XC with partial order (31) is not a lattice. Example 4. Any positive linear operator A on XC is a real operator; that is, A : X → X. In fact, let X ∋ x = x+ − x− . By definition, Ax+ ≥ 0 and Ax− ≥ 0 so Ax+ , Ax− ∈ X and thus Ax = Ax+ − Ax− ∈ X. Kinetic models in natural sciences 15 It is a more complicated task to introduce a norm on XC because standard product norms, in general, fail to preserve the homogeneity of the norm. Since XC is not a lattice, we cannot define the modulus of z = x+iy ∈ XC in a usual way. However, following an equivalent definition of the modulus in the scalar case, for x + iy ∈ XC we define |x + iy| = sup {x cos θ + y sin θ}. θ∈[0,2π] It can be proved that this element exists. Such a defined modulus has all standard properties of the scalar complex modulus. Thus, one can define a norm on the complexification XC by ∥z∥c = ∥x + iy∥c = ∥|x + iy|∥. (32) Properties (a)–(c) and |x| ≤ |z|, |y| ≤ |z| imply 1 (∥x∥ + ∥y∥) ≤ ∥z∥c ≤ ∥x∥ + ∥y∥, 2 hence ∥ · ∥c is a norm on XC which is equivalent to the Euclidean norm on X × X, denoted by ∥ · ∥C . As the norm ∥ · ∥ is a lattice norm on X, we have ∥z1 ∥c ≤ ∥z2 ∥c , whenever |z1 | ≤ |z2 |, and ∥ · ∥c becomes a lattice norm on XC . Definition 8. A complex Banach lattice is an ordered complex Banach space XC that arises as the complexification of a real Banach lattice X, according to Definition 7, equipped with the norm (32). Remember: a complex Banach lattice is not a Banach lattice! Any linear operator A on X can be extended to XC according to the formula AC (x + iy) = Ax + iAy. We observe that if A is a positive operator between real Banach lattices X and Y then, for z = x + iy ∈ XC , we have (Ax)cos θ + (Ay)sin θ = A(x cos θ + y sin θ) ≤ A|z|, therefore |AC z| ≤ A|z|. Hence for positive operators ∥AC ∥c = ∥A∥. (33) There are examples, where ∥A∥ < ∥AC ∥c . Note that the standard Lp (Ω) and C(Ω) norms are of the type (32). These spaces have a nice property of preserving the operator norm even for operators which are not necessarily positive, see [6, p. 63]. Remark 3. If for a linear operator A, we prove that it generates a semigroup of say, contractions, in X, then this semigroup will be also a semigroup of contractions on XC , hence, in particular, A is a dissipative operator in the complex setting. Due to this observation we confine ourselves to real operators in real spaces. 16 Jacek Banasiak Series of Positive Elements in Banach Lattices We note the following two results which are series counterparts of the dominated and monotone convergence theorems in Banach lattices. Theorem 3. Let (xn (t))n∈N be family of nonnegative sequences in a Banach lattice X, parameterized by a parameter t ∈ T ⊂ R, and let t0 ∈ T . (i) If for each n ∈ N the function t → xn (t) is non-decreasing and lim xn (t) = xn in norm, then t↗t0 lim ∞ ∑ xn (t) = t↗t0 n=0 ∞ ∑ xn , (34) n=0 irrespective of whether the right hand side exists in X or ∥ sup{∥ N ∑ ∞ ∑ xn ∥ := n=0 xn ∥; N ∈ N} = ∞. In the latter case the equality should be n=0 understood as the norms of both sides being infinite. (ii) If lim xn (t) = xn in norm for each n ∈ N and there exists (an )n∈N such t→t0 that xn (t) ≤ an for any t ∈ T, n ∈ N with ∞ ∑ ∥an ∥ < ∞, then (34) holds n=0 as well. ∑∞ Remark 4. Note that∑if X is a KB-space, then limt↗t0 n=0 xn (t) ∈ X im∞ plies convergence of n=0 xn . In fact, since xn ≥ 0 (by closedness of the pos∑N ∑∞ itive ∑ cone), N → n=0 xn is non-decreasing, and hence either∑ n=0 xn ∈ X, ∞ ∞ or ∥ n=0 xn ∥ = ∞, and in the latter case we have ∥limt↗t0 n=0 xn (t)∥ = ∞. 2.5 First semigroups As mentioned before, we are concerned with methods of finding solutions of the Cauchy problem: Definition 9. Given a complex or real Banach space and a linear operator A with domain D(A) and range RanA contained in X and also given an element ◦ ◦ u∈ X, find a function u(t) = u(t, u) such that 1. u(t) is continuous on [0, ∞[ and continuously differentiable on ]0, ∞[ in the norm of X, 2. for each t > 0, u(t) ∈ D(A) and ∂t u(t) = Au(t), t > 0, (35) 3. lim u(t) = u0 t→0+ in the norm of X. (36) Kinetic models in natural sciences 17 A function satisfying all conditions above is called the classical solution of (35), (36). If u(t) ∈ D(A) (and thus u ∈ C 1 ([0, ∞[, X)), then such a function is called a strict solution to (35), (36). To shorten notation, we denote by C k (I, X) a space of functions which, for each t ∈ I ⊂ R satisfy u(t) ∈ X and are continuously differentiable k times in t with respect to the norm of X. Thus, e.g. a classical solution u satisfies u ∈ C([0, ∞[, X) ∩ C 1 (]0, ∞[, X). Definitions and basic properties If the solution to (35), (36) is unique, then we can introduce the family of operators (G(t))t≥0 such that u(t, u0 ) = G(t)u0 . Ideally, G(t) should be defined on the whole space for each t > 0, and the function t → G(t)u0 should be continuous for each u0 ∈ X, leading to well-posedness of (35), (36). Moreover, uniqueness and linearity of A imply that G(t) are linear operators. A fine-tuning of these requirements leads to the following definition. Definition 10. A family (G(t))t≥0 of bounded linear operators on X is called a C0 -semigroup, or a strongly continuous semigroup, if (i) G(0) = I; (ii) G(t + s) = G(t)G(s) for all t, s ≥ 0; (iii) limt→0+ G(t)x = x for any x ∈ X. A linear operator A is called the (infinitesimal) generator of (G(t))t≥0 if Ax = lim+ h→0 G(h)x − x , h (37) with D(A) defined as the set of all x ∈ X for which this limit exists. Typically the semigroup generated by A is denoted by (GA (t))t≥0 . If (G(t))t≥0 is a C0 -semigroup, then the local boundedness and (ii) lead to the existence of constants M > 0 and ω such that for all t ≥ 0 ∥G(t)∥X ≤ M eωt (38) (see, e.g., [30, p. 4]). We say that A ∈ G(M, ω) if it generates (G(t))t≥0 satisfying (38). The type, or uniform growth bound, ω0 (G) of (G(t))t≥0 is defined as ω0 (G) = inf{ω; there is M such that (38) holds}. (39) Let (G(t))t≥0 be the semigroup generated by the operator A. From (37) and conditions (ii), (iii) of Definition 10, we see that if A is the generator of ◦ ◦ (G(t))t≥0 , then for u∈ D(A) the function t → G(t) u is a classical solution of the following Cauchy problem, 18 Jacek Banasiak ∂t u(t) = Au(t), t > 0, ◦ lim u(t) = u . t→0+ (40) (41) We note that ideally the generator A should coincide with A but in reality very often it is not so. However, for most of this chapter we are concerned with solvability of (40), (41); that is, with the case when A of (35) is the generator of a semigroup. ◦ We emphasize that, in general, the function u(t) = G(t) u is a classical so◦ ◦ lution to (40), (41) only for u∈ D(A). For u∈ X \ D(A), however, the function ◦ u(t) = G(t) u is continuous but, in general, not differentiable. Nevertheless, ∫t it follows that v(t) = 0 u(s)ds ∈ D(A) and u satisfies the integrated version of (40), (41): ∫t ◦ u(t) = A u(s)ds+ u . (42) 0 We say that a function u satisfying (??) (or, equivalently, (42)) is a mild solution or integral solution of (40), (41). Thus, if we have a semigroup, we can identify the Cauchy problem of which it is a solution. Usually, however, we are interested in the reverse question, that is, in finding the semigroup for a given equation. The answer is given by the Hille–Yoshida theorem (or, more properly, the Feller–Miyadera–Hille– Phillips–Yosida theorem). Before, however, we need to recall some terminology related to the spectrum of an operator. Interlude - the spectrum of an operator Let us recall that the resolvent set of A is defined by ρ(A) = {λ ∈ C; (λ − A)−1 ∈ L(X)} and, for λ ∈ ρ(A), we define the resolvent of A by R(λ, A) = (λI − A)−1 . The complement of ρ(A) in C is called the spectrum of A and denoted by σ(A). In general, it is possible that either ρ(A) or σ(A) is empty. The spectrum is usually subdivided into several subsets. • • • Point spectrum σp (A) is the set of λ ∈ σ(A) for which the operator λI − A is not one-to-one. In other words, σp (A) is the set of all eigenvalues of A. Residual spectrum σr (A) is the set of λ ∈ σ(A) for which λI − A is oneto-one but Ran (λI − A) is not dense in X. Continuous spectrum σc (A) which is the set of λ ∈ σ(A) for which the operator λI − A is one-to-one and its range is dense in, but not equal to, X Kinetic models in natural sciences 19 The resolvent of any operator A satisfies the resolvent identity R(λ, A) − R(µ, A) = (µ − λ)R(λ, A)R(µ, A), λ, µ ∈ ρ(A). (43) For any bounded operator, the spectrum is a compact subset of C so that ρ(A) ̸= ∅. If A is bounded, then the limit √ r(A) = lim n ∥An ∥ (44) n→∞ exists and is called the spectral radius. Clearly, r(A) ≤ ∥A∥. Equivalently, r(A) = sup |λ|. (45) λ∈σ(A) For an unbounded operator A the role of the spectral radius often is played by the spectral bound s(A) defined as s(A) = sup{ℜλ; λ ∈ σ(A)}. (46) Hille-Yosida theorem Theorem 4. A ∈ G(M, ω) if and only if (a) A is closed and densely defined, (b) there exist M > 0, ω ∈ R such that (ω, ∞) ⊂ ρ(A) and for all n ≥ 1, λ > ω, M ∥(λI − A)−n ∥ ≤ . (47) (λ − ω)n If A is the generator of (G(t))t≥0 then, for any x ∈ X, ℜλ > ω ∫∞ R(λ, A)x = e−λt G(t)xdt. (48) 0 A widely used approximation formula, which can also be used in the generation proof, is the operator version of the well-known scalar formula: if A is the generator of a C0 -semigroup (G(t))t≥0 , then for any x ∈ X, ( )−n ( n ( n ))n t I− A x = lim R ,A x n→∞ n→∞ t n t G(t)x = lim and the limit is uniform in t on bounded intervals, [30, Theorem 1.8.3]. (49) 20 Jacek Banasiak Dissipative operators and contractive semigroups Let X be a Banach space (real or complex) and X ∗ be its dual. From the Hahn–Banach theorem, for every u ∈ X there exists u∗ ∈ X ∗ satisfying <u∗ , u>= ∥u∥2 = ∥u∗ ∥2 . Therefore the duality set J (u) = {u∗ ∈ X ∗ ; <u∗ , u>= ∥u∥2 = ∥u∗ ∥2 } (50) is nonempty for every u ∈ X. Definition 11. We say that an operator (A, D(A)) is dissipative if for every u ∈ D(A) there is u∗ ∈ J (u) such that ℜ <u∗ , Ax> ≤ 0. (51) An important equivalent characterisation of dissipative operators, [30, Theorem 1.4.2], is that A is dissipative if and only if for all λ > 0 and u ∈ D(A), ∥(λI − A)u∥ ≥ λ∥u∥. (52) Combination of the Hille–Yosida theorem with the above property gives a generation theorem for dissipative operators, known as the Lumer–Phillips theorem ([30, Theorem 1.43] or [20, Theorem II.3.15]). Theorem 5. For a densely defined dissipative operator (A, D(A)) on a Banach space X, the following statements are equivalent. (a) The closure A generates a semigroup of contractions. (b) Ran(λI − A) = X for some (and hence all) λ > 0. If either condition is satisfied, then A satisfies (51) for any u∗ ∈ J (u). In other words, to prove that (the closure of) a dissipative operators generates a semigroup, we only need to show that the equation λu − Au = f (53) is solvable for f from (a dense subset of) X for some λ > 0. We do not need to prove that the solution is a resolvent satisfying (47). In particular, if we know that A is closed, then the density of Ran(λI − A) is sufficient for A to be a generator. On the other hand, if we do not know a priori that A is closed, then Ran(λI − A) = X yields A being closed and consequently that it is the generator. Example 5. Let us have a look at the classical problem which often is incorrectly solved. Consider Au = −∂x u, x ∈ (0, 1), Kinetic models in natural sciences 21 on D(A) = {u ∈ W11 (I); u(0) = 0}, where I =]0, 1[. The state space is real X = L1 (I). For a given u ∈ X, we have { ∥u∥signu(x) if u(x) ̸= 0, J (u) = α ∈ [−∥u∥, ∥u∥] if u(x) = 0. Note that J is a multivalued function. Further, by, say, [15], any element of W11 (I) can be represented by an absolutely continuous (and thus continuous) function on I. Now, for v ∈ J (u) we have ∫1 <−∂x u, v> = −∥u∥ ∂x u(x)signu(x)dx 0 ∫ = −∥u∥ ∫ ∂x u(x)dx − {x∈I; u(x)>0} ∂x u(x)dx . {x∈I; u(x)<0} Since u is continuous, both sets I+ := {x ∈ I; u(x) > 0} and I− := {x ∈ I; u(x) < 0} are open. Then, see [2, p. 42], ∑ I± = ]αn± , βn± [ n where Then ]αn± , βn± [ are non overlapping open intervals and the sum is countable. ∫ ∂x u(x)dx = ∑ (u(βn± ) − u(αn± )) = { n I± u(1) if 1 ∈ I± 0 if 1 ∈ / I± as 1 only can be the right end of the component intervals and we used u(0) = 0. Now, if 1 ∈ I+ , then u(1) > 0, if 1 ∈ I− , then u(1) < 0, and if 1 ∈ / I+ ∪ I− , then u(1) = 0. In any case, <−∂x u, v>≤ 0 and the operator (A, D(A)) is dissipative. Clearly, the solution of λu + ∂x u = f, is given by u(x) = e−λx ∫x eλs f (s)ds and, for λ > 0, 0 ∫1 ∥u∥ ≤ 0 u(0) = 0, e−λx ∫x eλs |f (s)|ds dx ≤ ∥f ∥ 0 which gives solvability of (53). We note that, of course, with a more careful integration we would be able to obtain the Hille-Yosida estimate (47). This additional work is, however, not necessary for dissipative operators. 22 Jacek Banasiak Analytic semigroups In the previous paragraph we noted that if an operator is dissipative, then we can prove that it generates a semigroup provided (53) is solvable. It turns out that the solvability of (53) can be used to prove that A generates a semigroup without assuming that it is dissipative, but then we must consider complex λ. Hence, let the inverse (λI − A)−1 exists in the sector π S π2 +δ := {λ ∈ C; |arg λ| < + δ} ∪ {0}, (54) 2 for some 0 < δ < π2 , and let there exist C such that for every 0 ̸= λ ∈ S π2 +δ the following estimate holds: ∥R(λ, A)∥ ≤ C . |λ| (55) Then A is the generator of a uniformly bounded semigroup (GA (t))t≥0 (the constant M in (38) not necessarily equals C) and (A(t))t≥0 is given by the integral ∫ 1 GA (t) = eλt R(λ, A)dλ, (56) 2πi Γ where Γ is an unbounded smooth curve in S π2 +δ . The reason why (A(t))t≥0 is called an analytic semigroup is that it extends to an analytic function on Sδ . The estimate (55) is sometimes awkward to prove as it requires the knowledge of the resolvent in the whole sector. The result given in [20, Theorem II 4.6] allows for restriction of the estimates to a positive half plane. Theorem 6. An operator (A, D(A)) on a Banach space X generates a bounded analytic semigroup (GA (z))z∈Sδ in a sector Sδ if and only if A generates a bounded strongly continuous semigroup (GA (t))t≥0 and there exists a constant C > 0 such that C ∥R(r + is, A)∥ ≤ (57) |s| for all r > 0 and 0 ̸= s ∈ R. This result can be generalized to arbitrary analytic semigroups: (A, D(A)) generates an analytic semigroup (GA (z))z∈Sδ if and only if A generates a strongly continuous semigroup (GA (t))t≥0 and there exist constants C > 0, ω > 0 such that C ∥R(r + is, A)∥ ≤ (58) |s| for all r > ω and 0 ̸= s ∈ R. If A is the generator of an analytic semigroup (GA (t))t≥0 , then t → GA (t) ◦ has derivatives of arbitrary order on ]0, ∞[. This shows that t → GA (t) u ◦ solves the Cauchy problem (36) for arbitrary u ∈ X. This is a significant ◦ improvement upon the case of C0 -semigroup, for which u ∈ D(A) was required. Kinetic models in natural sciences 23 Fractional powers of generators and interpolation spaces If A generates an analytic semigroup, the formula (56) can be regarded as the extension of the definition of etA via the so-called Dunford integral type functional calculus, [30]. In a similar way we can define f (A) where f is any scalar function which is analytic in an open neighbourhood of the spectrum of A and such that the integral (56) is convergent. One of the most important choices is f (λ) = λ−α , where λ−α is real for λ > 0 and has a cut along the negative real axis. This gives rise to bounded operators (−A)−α defined by ∫ 1 (−A)−α = λ−α (A − λI)−1 dλ, (59) 2πi Γ The change of sign is dictated by the fact that changing by A to (−A), we also change the position of the spectrum with respect to the integration curve. Provided 0 ∈ / ρ(−A), by inversion, we define unbounded operators (−A)α . We denote by D((−A)α ) the domain of (−A)α . It follows that D(A) ⊂ D((−A)α ) ⊂ X if 0 < α < 1. For example, if A = ∆ on the maximal domain in L2 (Rn ), then D(A) = W22 (Rn ) and D((−A)α ) = W22α (Rn ). We note an important property of fractional powers of generators and of corresponding analytic semigroups, which will used in the sequel. If (A(t))t≥0 is an analytic semigroup, then for every t > 0 and α ≥ 0, we have (−A)α GA (t) = GA (t)(−A)α , the operator (−A)α GA (t) is bounded and ∥tα (−A)α GA (t)∥ ≤ Mα (60) for some constant Mα . Example 6. Consider the simple example which will be useful later. Let X = L1 ([0, ∞[) and [Au] = −a(x)u(x) where a(x) ≥ a0 > 0. Then −Au = au and ρ(−A) = {λ ; λ ∈ / Ran a}. Using the definition, we find ∫ 1 λ−α (−A)−α = dλ, 2πi a(x) − λ Γ where x is a parameter. The integral can be evaluated by closing the contour by an arc running in the positive half-plane and enclosing the only pole of the integrand at λ = a(x). Thus, noting the now the contour orientation is clockwise and using the Cauchy theorem, we find [(−A)−α u](x) = a−α (x)u(x). 24 Jacek Banasiak The spaces D((−A)α ) serve a an important class of intermediate spaces between D(A) and X. However, in some situations they are not sufficient. Formula (60) can be written as ∥t1−α (−A)1−α GA (t)(−A)α x∥ ∈ L∞ (]0, 1[) whenever x ∈ D((−A)α ) which could serve as a characterization of D((−A)α ). Taking this as a starting point, let (A, D(A)) be the generator of an analytic semigroup (GA (t))t≥0 on a Banach space X. Then we construct a family of intermediate spaces, DA (α, r), 0 < α < 1, 1 ≤ r ≤ ∞ in the following way: DA (α, r) := {x ∈ X : t → v(t) := ∥t1−α−1/r AGA (t)x∥X ∈ Lr (I)}, (61) ∥x∥DA (α,r) := ∥x∥X + ∥v(t)∥Lr (I) , (62) where I := (0, 1); see [27, p.45]. It turns out that these spaces can be identified with real interpolation spaces between X and D(A) and one can use a rich theory of the latter. In particular, by [27, Corollary 2.2.3], these spaces do not depend explicitly on A, but only on D(A) and its graph norm. This is in contrast to D((−A)α ) which only satisfy DA (α, 1) ⊂ D((−A)α ) ⊂ DA (α, ∞) (63) and may depend on the particular form of the operator A. Which makes the spaces DA (α, r) as useful as D((−A)α ) in dealing with the semigroup generated by A is the fact that ∥R(λ, A)x∥DA (α,r) = ∥R(λ, A)x∥X + = ∥R(λ, A)x∥L(X) + ∫1 1/r ∥s1−α−1/r AGA (s)R(λ, A)x∥rX 0 1/r ∫1 ∥R(λ, A)(s 1−α−1/r 0 ≤ ∥R(λ, A)∥L(X) ∥x∥X + ∫1 AGA (s)x)∥rX ds 1/r ∥(s1−α−1/r AGA (s)x)∥rX ds 0 ≤ ∥R(λ, A)∥L(X) ∥x∥DA (α,r) . This leads to the following observation. Proposition 2. Let Aα,r be the part of A in DA (α, r). Then ρ(Aα,r ) ⊂ ρ(A), ∥R(λ, Aα,r ∥L(DA (α,r)) ≤ ∥R(λ, A)∥L(X) for λ ∈ ρ(A). Consequently, Aα,r generates an analytic semigroup in DA (α, r). Kinetic models in natural sciences 25 Nonhomogeneous Problems Let us consider the problem of finding the solution to the Cauchy problem: ∂t u = Au + f (t), 0 < t < T, ◦ u(0) = u, (64) where 0 < T ≤ ∞, A is the generator of a semigroup and f : (0, T ) → X is a known function. If we are interested in classical solutions, then clearly f must be continuous. However, this condition proves to be insufficient. We thus generalise the concept of the mild solution introduced in (42). We observe that if u is a classical solution of (64), then it must be given by ◦ ∫t G(t − s)f (s)ds u(t) = G(t) u + (65) 0 (see, e.g., [30, Corollary 4.2.2]). The integral is well defined even if f ∈ ◦ L1 ([0, T ], X) and u∈ X. We call u defined by (65) the mild solution of (64). For an integrable f such u is continuous but not necessarily differentiable, and therefore it may be not a solution to (64). We have the following theorem giving sufficient conditions for a mild solution to be a classical solution (see, e.g., [30, Corollary 4.2.5 and 4.2.6]). Theorem 7. Let A be the generator of a C0 -semigroup (GA (t))t≥0 and x ∈ D(A). Then (65) is a classical solution of (64) if either (i) f ∈ C 1 ([0, T ], X), or (ii) f ∈ C([0, T ], X) ∩ L1 ([0, T ], D(A)). If the semigroup (GA (t))t≥0 generated by A is analytic, then the requirements imposed on f can be substantially weakened. We have then the following counterpart of Theorem 7. Theorem 8. Let A be the generator of an analytic semigroup (GA (t))t≥0 , ◦ u∈ X and f ∈ L1 ([0, T ], X). Then (65) is the classical solution of (64) if either (i) f is locally Hölder continuous on ]0, T [, or (ii) ∃{α > 0} f ∈ C(]0, T ], X)∩L1 ([0, T ], D((−A)α )) and t → ∥(−A)α f (t)∥X is bounded over compact subsets of ]0, T ]. Part (ii) of this theorem has been proved in [9]. An important refinement of Theorem 8 which becomes very useful in nonlinear problems is that, actually, the solution has a better regularity. In fact, under assumption (i) we additionally have 0,1−r u ∈ Cloc (]0, T [, D((−A)r )), 0 ≤ r < 1. (66) By (63), the statement of the above theorem holds if the domains of the fractional power are replaced by an appropriate D(α, 1). 26 Jacek Banasiak Positive Semigroups Definition 12. Let X be a Banach lattice. We say that the semigroup (G(t))t≥0 on X is positive if for any x ∈ X+ and t ≥ 0, G(t)x ≥ 0. We say that an operator (A, D(A)) is resolvent positive if there is ω such that (ω, ∞) ⊂ ρ(A) and R(λ, A) ≥ 0 for all λ > ω. Remark 5. In this section, because we address several problems related to spectral theory, we need complex Banach lattices. Let us recall, Definitions 7 and 8, that a complex Banach lattice is always a complexification XC of an underlying real Banach lattice X. In particular, x ≥ 0 in XC if and only if x ∈ X and x ≥ 0 in X. It is easy to see that a strongly continuous semigroup is positive if and only if its generator is resolvent positive. In fact, the positivity of the resolvent for λ > ω follows from (48) and closedness of the positive cone; see Proposition 1. Conversely, the latter with the exponential formula (49) shows that resolvent positive generators generate positive semigroups. There are several results on the resolvent of a positive semigroup which we collect in the following theorem. Theorem 9. [29, Theorem 1.4.1] Let (GA (t))t≥0 be a positive semigroup on a Banach lattice, with generator A. Then ∫∞ R(λ, A)x = e−λt GA (t)xdt (67) 0 for all λ ∈ C with ℜλ > s(A). Furthermore, (i) Either s(A) = −∞ or s(A) ∈ σ(A); (ii) For a given λ ∈ ρ(A), we have R(λ, A) ≥ 0 if and only if λ > s(A); (iii) For all ℜλ > s(A) and x ∈ X, we have |R(λ, A)x| ≤ R(ℜλ, A)|x|. We conclude this section by briefly describing an approach of [3] which has a number of interesting applications. To fix attention, assume for the time being that ω < 0 (thus, in particular, A is invertible and −A−1 = R(0, A)) and λ > 0. The resolvent identity −A−1 = (λI − A)−1 + λ(λI − A)−1 (−A−1 ), can be extended by induction to −A−1 = R(λ, A) + λR(λ, A)2 + · · · + λn R(λ, A)n (−A−1 ). Now, because all terms above are nonnegative, we obtain (68) Kinetic models in natural sciences 27 sup {λn ∥(λ − A)−n (−A−1 )∥X } = M < +∞. n∈N,λ>ω This is ‘almost’ the Hille–Yosida estimate and allows us to prove that the Cauchy problem (40), (41) has a mild Lipschitz continuous solution for ◦ u∈ D(A2 ). If, in addition, A is densely defined, then this mild solution is differentiable, and thus it is a strict solution (see, e.g., [4] and [5, pp. 191– 200]). These results are obtained by means of the integrated, or regularised, semigroups, which are beyond the scope of this lecture, so we do not enter into details of this very rich field. We mention, however, an interesting consequence of (68) for semigroup generation which has already found several applications and which we use later. Theorem 10. [3, 14] Let A be a densely defined resolvent positive operator. If there exist λ0 > s(A), c > 0 such that for all x ≥ 0, ∥R(λ0 , A)x∥X ≥ c∥x∥X , (69) then A generates a positive semigroup (GA (t))t≥0 on X. Proof. Let us take s(A) < ω ≤ λ0 and set B = A − ωI so that s(B) < 0. Because R(0, B) = R(ω, A) ≥ R(λ0 , A), it follows from (69) and (30) that ∥R(0, B)x∥X ≥ ∥R(λ0 , A)x∥X ≥ c∥x∥X for x ≥ 0. Using (68) for B and taking x = λn R(λ, B)n g, g ≥ 0, we obtain, by (69), ∥λn R(λ, B)n g∥X ≤ c−1 ∥R(0, B)λn R(λ, B)n g∥ ≤ c−1 ∥R(0, B)g∥X ≤ M ∥g∥X , for λ > 0. Again, by (30), we can extend the above estimate onto X proving the Hille–Yosida estimate. Because B is densely defined, it generates a bounded positive semigroup and thus ∥GA (t)∥ ≤ M eωt . ⊓ ⊔ Perturbation techniques Verifying conditions of the Hille–Yosida, or even the Lumer–Phillips, theorems for a concrete problem is quite often a formidable task. On the other hand, in many cases the operator appearing in the evolution equation at hand is built as a combination of much simpler operators that are relatively easy to analyse. The question now is to what extent the properties of these simpler operators are inherited by the full equation. More precisely, we are interested in the problem: Problem P. Let (A, D(A)) be a generator of a C0 -semigroup on a Banach space X and (B, D(B)) be another operator in X. Under what conditions does A + B, or an extension K of A + B, generate a C0 semigroup on X? 28 Jacek Banasiak We note that the situation when K = A + B is quite rare. Usually at best we can show that there is an extension of A+B (another realization of K = A+B) which is the generator. The reason for this is that, unless B is in some sense strictly subordinated to A, adding B to A may significantly alter some vital properties of A. The identification of K in such cases usually is a formidable task. A Spectral Criterion Usually the first step in establishing whether A + B, or some of its extensions, generates a semigroup is to find if λI − (A + B) (or its extension) is invertible for all sufficiently large λ. In all cases discussed here we have the generator (A, D(A)) of a semigroup and a perturbing operator (B, D(B)) with D(A) ⊆ D(B). We note that B is A-bounded; that is, for some a, b ≥ 0 we have ∥Bx∥ ≤ a∥Ax∥ + b∥x∥, x ∈ D(A) (70) if and only if BR(λ, A) ∈ L(X) for λ ∈ ρ(A). In what follows we denote by K an extension of A + B. We now present an elegant result relating the invertibility properties of λI − K to the properties of 1 as an element of the spectrum of BR(λ, A), first derived in [?]. Theorem 11. Assume that Λ = ρ(A) ∩ ρ(K) ̸= ∅. (a) 1 ∈ / σp (BR(λ, A)) for any λ ∈ Λ; (b) 1 ∈ ρ(BR(λ, A)) for some/all λ ∈ Λ if and only if D(K) = D(A) and K = A + B; (c) 1 ∈ σc (BR(λ, A)) for some/all λ ∈ Λ if and only if D(A) K = A + B; D(K) and (d) 1 ∈ σr (BR(λ, A)) for some/all λ ∈ Λ if and only if K ) A + B. Corollary 1. Under the assumptions of Theorem 11, K = A + B if one of the following criteria is satisfied: for some λ ∈ ρ(A) either (i) BR(λ, A) is compact (or, if X = L1 (Ω, dµ), weakly compact), or (ii) the spectral radius r(BR(λ, A)) < 1. Proof. If (ii) holds, then obviously I − BR(λ, A) is invertible by the Neumann series theorem: (I − BR(λ, A))−1 = ∞ ∑ (BR(λ, A))n , n=0 giving the thesis by Proposition 11 (b). Additionally, we obtain (71) Kinetic models in natural sciences R(λ, A + B) = R(λ, A)(I − BR(λ, A))−1 = R(λ, A) ∞ ∑ 29 (BR(λ, A))n . (72) n=0 If (i) holds, then either BR(λ, A) is compact or, in L1 setting, (BR(λ, A))2 is compact, [19, p. 510], and therefore, if I − BR(λ, A) is not invertible, then 1 must be an eigenvalue, which is impossible by Theorem 11(c). ⊓ ⊔ If we write the resolvent equation (λI − (A + B))x = y, y ∈ X, (73) in the (formally) equivalent form x − R(λ, A)Bx = R(λ, A)y, (74) then we see that we can hope to recover x provided the Neumann series R(λ)y := ∞ ∑ (R(λ, A)B)n R(λ, A)y = n=0 ∞ ∑ R(λ, A)(BR(λ, A))n y. (75) n=0 is convergent. Clearly, if (71) converges, then we can factor out R(λ, A) from the series above getting again (72). However, R(λ, A) inside acts as a regularising factor and (75) converges under weaker assumptions than (71) and this fact is frequently used to construct the resolvent of an extension of A + B (see, e.g., Theorem 15). The most often used perturbation theorem is the Bounded Perturbation Theorem and Related Results, see e.g. [20, Theorem III.1.3] Theorem 12. Let (A, D(A)) ∈ G(M, ω) for some ω ∈ R, M ≥ 1. If B ∈ L(X), then (K, D(K)) = (A + B, D(A)) ∈ G(M, ω + M ∥B∥). In many cases the Bounded Perturbation Theorem gives insufficient information. It can be combined with the Trotter product formula, [20, 30]. Assume K0 is of type (1, ω0 ), and K1 is of type (1, ω1 ), ω0 , ω1 ∈ R,. If we know that (K, D(K0 ) ∩ D(K1 )) := (K0 + K1 , D(K0 ) ∩ D(K1 )) generates a semigroup, then n GK (t)x = lim (GK0 (t/n)GK1 (t/n)) x, n→∞ x ∈ X, (76) uniformly in t on compact intervals and K is of type (1, ω) with ω = ω0 + ω1 . Moreover, if both semigroups (GK0 (t))t≥0 and (GK1 (t))t≥0 are positive, then (GK (t))t≥0 is positive. The assumption of boundedness of B, however, is often too restrictive. Another frequently used result uses special structure of dissipative operators. Theorem 13. Let A and B be linear operators in X with D(A) ⊆ D(B) and A + tB is dissipative for all 0 ≤ t ≤ 1. If ∥Bx∥ ≤ a∥Ax∥ + b∥x∥, (77) for all x ∈ D(A) with 0 ≤ a < 1 and for some t0 ∈ [0, 1] the operator (A+t0 B, D(A)) generates a semigroup (of contractions), then A+tB generates a semigroup of contractions for every t ∈ [0, 1]. 30 Jacek Banasiak 2.6 Positive perturbations of positive semigroups Perturbation results can be significantly strengthened in the framework of positive semigroups. We have seen in (71) that the condition r(BR(λ, A)) < 1 implies invertibility of λI − (A + B). It turns out that this condition is equivalent to invertibility for positive perturbations of resolvent positive operators. Theorem 14. Assume that X is a Banach lattice. Let A be a resolvent positive operator in X and λ > s(A). Let B : D(A) → X be a positive operator. Then the following are equivalent, (a) r(B(λI − A)−1 ) < 1; (b) λ ∈ ρ(A + B) and (λI − (A + B))−1 ≥ 0. If either condition is satisfied, then (λI − A − B)−1 = (λI − A)−1 ∞ ∑ (B(λI − A − B)−1 )n ≥ (λI − A)−1 . (78) n=0 Kato-Voigt type results Here we consider only X = L1 (Ω, dµ) where (Ω, µ) is a measure space. We recall that if Z ⊂ X is a subspace, then by Z+ we denote the cone of nonnegative elements of Z and for f ∈ X, the symbols f± denote the positive and negative part of f ; that is, f+ = max{f, 0} and f− = − min{f, 0}. Let (G(t))t≥0 be a strongly continuous semigroup on X. We say that (G(t))t≥0 is a substochastic semigroup if for any t ≥ 0 and x ≥ 0, G(t)x ≥ 0 and ∥G(t)x∥ ≤ ∥x∥, and a stochastic semigroup if additionally ∥G(t)f ∥ = ∥f ∥ for f ∈ X+ . Theorem 15. Let X = L1 (Ω) and suppose that the operators A and B satisfy 1. (A, D(A)) generates a substochastic semigroup (GA (t))t≥0 ; 2. D(B) ⊃ D(A) and Bu ≥ 0 for u ∈ D(B)+ ; 3. for all u ∈ D(A)+ ∫ (Au + Bu)dµ ≤ 0. (79) Ω Then there is an extension (K, D(K)) of (A + B, D(A)) generating a C0 semigroup of contractions, say, (GK (t))t≥0 . The generator K satisfies R(λ, K)u = ∞ ∑ R(λ, A)(BR(λ, A))k u, k=0 λ > 0. (80) Kinetic models in natural sciences 31 Proof. First, assumption (79) gives us dissipativity on the positive cone. Next, let us take u = R(λ, A)x = (λI − A)−1 x for x ∈ X+ so that u ∈ D(A)+ . Because R(λ, A) is a surjection from X onto D(A), by (A + B)u = (A + B)R(λ, A)x = −x + BR(λ, A)x + λR(λ, A)x, we have ∫ − ∫ x dµ + Ω ∫ R(λ, A)x dµ ≤ 0. BR(λ, A)x dµ + λ Ω (81) Ω Rewriting the above in terms of the norms, we obtain λ∥R(λ, A)x∥ + ∥BR(λ, A)x∥ − ∥x∥ ≤ 0, x ∈ X+ , (82) from which ∥BR(λ, A)∥ ≤ 1. We define operators Kr , 0 ≤ r < 1 by Kr = A + rB, D(Kr ) = D(A). We see that the spectral radius of rBR(λ, A) does not exceed r < 1, the resolvent (λI − (A + rB))−1 exists and is given by R(λ, Kr ) := (λI − (A + rB))−1 = R(λ, A) ∞ ∑ n rn (BR(λ, A)) , (83) n=0 where the series converges absolutely and each term is positive. Let x ∈ D(A)+ . Then we have, for r < 1, ∫ ∫ (A + B)xdµ + (r − 1) (A + rB)xdµ, = Ω ∫ Ω Bxdµ ≤ 0 (84) Ω by assumptions. Thus, by the above, ∫ ∥(λI − Kr )x∥ ≥ ∫ (λI − Kr )xdµ = λ∥x∥ − Ω Kr xdµ ≥ λ∥x∥, Ω for all x ∈ D(A)+ , by (84). Hence, ∥R(λ, Kr )y∥ ≤ λ−1 ∥y∥ (85) for all y ∈ X+ and, because R(λ, Kr ) is positive, (85) can be extended to the whole space X, by (29). Therefore, by the Lumer–Phillips theorem, for each 0 ≤ r < 1, (Kr , D(A)) generates a contraction semigroup which we denote (Gr (t))t≥0 . Since (R(λ, Kr )x)0≤r<1 is increasing as r ↑ 1 for each x ∈ X+ , and {∥R(λ, Kr )x∥}0≤r<1 is bounded and X = L1 (Ω) is a KB-space, there is an element yλ,x ∈ X+ such that lim R(λ, Kr )x = yλ,x r→1− 32 Jacek Banasiak in X. By the Banach–Steinhaus theorem we obtain the existence of a bounded positive operator on X, denoted by R(λ), such that R(λ)x = yλ,x . We use the Trotter–Kato theorem to obtain that R(λ) is defined for all λ > 0 and it is the resolvent of a densely defined closed operator K which generates a semigroup of contractions (GK (t))t≥0 . Moreover, for any x ∈ X, lim Gr (t)x = GK (t)x, (86) r→1− and the limit is uniform in t on bounded intervals and, provided x ≥ 0, monotone as r ↑ 1. By the monotone convergence theorem, Theorem 3, R(λ, K)x = ∞ ∑ R(λ, A)(BR(λ, A))k x, x∈X (87) k=0 and we can prove that R(λ, K)(λI − (A + B))x = x which shows that K ⊇ A + B. ⊓ ⊔ The identification of K is a much more difficult task. We note that (79) can be written as ∫ (Au + Bu) dµ = −c(u) ≤ 0, (88) Ω where c is a positive functional on D(T ). We assume that c has the monotone convergence property: c(un ) → cu for un ↑ u, then cu = c(u) (for instance, it is an integral functional). The following theorem is fundamental for characterizing the generator of the semigroup. Theorem 16. For any fixed λ > 0, there is 0 ≤ βλ ∈ X ∗ with ∥βλ ∥ ≤ 1 such that for any f ∈ X+ , λ∥R(λ, K)f ∥ = ∥f ∥− <βλ , f> − c (R(λ, K)f ) . (89) In particular, c extends to a nonnegative continuous linear functional on D(K). It turns out that (BR(λ, T ))∗ βλ = βλ . (90) and hence, if βλ ̸= 0, then σp (BR(λ, T ))∗ ̸= ∅. This implies that σr (BR(λ, T )) ̸= ∅ and, by Theorem 11 d), K ̸= A + B. Another result, though not as elegant, is often more useful. It is based on the observation that the following are equivalent: (a) K = T + B. ∫ (b) Ku dµ ≥ −c(u), u ∈ D(K)+ . Ω Though the implication (b) ⇒ (a) seems to be useless as it require the knowledge of K which is what we are looking for, we note that if we can prove it for an extension of K, then obviously it will be valid for K. Hence Kinetic models in natural sciences Theorem 17. [6] If there exists an extension K such that for all u ∈ D(K)+ , then K = A + B. ∫ Ω 33 Ku dµ ≥ −c(u) Often one can use Kmax as the required extension of K. Arendt-Rhandi theorem Theorem 18. Assume that X is a Banach lattice, (A, D(A)) is a resolvent positive operator which generates an analytic semigroup and (B, D(A)) is a positive operator. If (λ0 I − (A + B), D(A)) has a nonnegative inverse for some λ0 larger than the spectral bound s(A) of A, then (A + B, D(A)) generates a positive analytic semigroup. Proof. The proof is an application of Theorem 14. Under assumptions of this theorem, we obtain that r(BR(λ0 , A)) < 1. In particular, the series ∞ ∑ (BR(λ0 , A))n converges in the uniform operator topology. Next, by Then=0 orem 9, R(λ, A) ≥ 0 if and only if ρ(A) ∋ λ > s(A). Thus, using the resolvent identity we have R(λ, A) = R(λ0 , A) − (λ − λ0 )R(λ0 , A)R(λ, A) ≤ R(λ0 , A) whenever λ ≥ λ0 . Since BR(λ, A) is bounded in X, see Theorem 1, B : D(A) → X is bounded in the graph norm of D(A). Let us now take λ ∈ C with ℜλ ≥ λ0 , R ∋ µ > λ0 and f ∈ D(A). Then µR(µ, A)R(λ, A)f → R(λ, A)f as µ → ∞ in the graph norm of D(A), see e.g., [30, Lemmas 1.3.2 and 1.3.3] and we have, for f ∈ D(A) |BR(λ, A)f | = lim |BR(µ, A)R(λ, A)| ≤ lim BR(µ, A)R(ℜλ, A)|f | µ→∞ µ→∞ = BR(µ, A)R(ℜλ, A)|f |, where we used |R(λ, A)f | ≤ R(ℜλ, A)|f | for ℜλ > s(A), see Theorem 9. Thus, by density, |BR(λ, A)f | ≤ BR(ℜλ, A)|f | (91) for all f ∈ X and therefore r(BR(λ, A)) ≤ r(BR(λ0 , A)) < 1 for any λ ∈ C with ℜλ ≥ λ0 . In particular, ∞ ∑ (BR(λ, A))n converges to a n=0 bounded linear operator with ∞ ∞ ∑ ∑ n n ≤ Mλ0 ∥f ∥, (BR(λ, A)) f ≤ (BR(λ , A)) |f | 0 n=0 n=0 uniformly for λ ∈ C with ℜλ > λ0 . 34 Jacek Banasiak Next we consider analyticity issue. Using Theorem 6 for the operator A, there are ωA and MA such that ∥R(r + is, A)∥ ≤ MA |s| for r > ωA . Taking now ω > max{λ0 , ωA } we have, by (78), ∞ ∞ MA ∑ ∑ n n ∥R(r + is, A + B)f ∥ = R(λ, A) (BR(λ, A)) f ≤ (BR(λ, A)) f |s| n=0 n=0 MA Mλ0 ∥f ∥, f ∈ X, ≤ |s| for all r > ω. Therefore (A + B, D(A)) generates an analytic semigroup. ⊓ ⊔ 2.7 Semi-linear problems Next we consider the semilinear abstract Cauchy problem ∂t u = Au + f (t, u), t > 0, ◦ u(0) = u, (92) where A is a generator of a C0 -semigroup (GA (t))t≥0 and f : [0, T ] × X → X is a known function. Since a priori we don’t know properties of the solution u (which may even fail to exist), it is plausible to start from a weaker formulation of the problem i.e. from the integral equation: ◦ ∫t GA (t − s)f (s, u(s))ds. u(t) = GA (t) u + (93) 0 This form is typical for fixed point techniques. We shall focus on the Banach contraction principle which leads to Theorem 19 below. It requires a relatively strong regularity from f as in the following definition. We say that f : [0, T ] × X → X is locally Lipschitz continuous in u, uniformly in t on bounded intervals if ∀{t′ ∈ [0, T [, c > 0}∃{L(c, t′ )}∀{t ∈ [0, t′ ], ∥u∥, ∥v∥ ≤ c} such that ∥f (t, u) − f (t, u)∥X ≤ L(c, t′ )∥u − v∥X Theorem 19. Let f : [0, ∞[×X → X be continuous in t, for t ∈ [0, ∞[ and locally Lipschitz continuous in u, uniformly in t, on bounded intervals. If A is the generator of a C0 -semigroup (GA (t))t≥0 on X then ∀u∈X ∃tmax >0 such ◦ that the problem (93) has a unique mild solution u on [0, tmax [ . Moreover, if tmax < +∞, then lim ∥u(t)∥X = ∞. t→∞ Kinetic models in natural sciences 35 The proof is done by Picard iterations as in the scalar case. Also, similarly to the scalar case, a sufficient condition for the existence of a global mild solution is that f be uniformly Lipschitz continuous on X. Uniform Lipschitz continuity yields at most linear growth in ∥x∥ of ∥f (t, x)∥. In fact, even for f (u) = u2 and A = 0, the blow-up occurs in finite time if the initial data are sufficiently large. There are two standard sufficient conditions ensuring that the mild solution, described in Theorem 19, is a classical solution. Both follow from the corresponding results for nonhomogeneous problems. They are either that f : [0, ∞[×X → X is continuously differentiable with respect to both variables, or that f : [0, ∞[×D(A) → D(A) is continuous. Certainly, in both cases to ensure that the solution is a classical solution we must assume that ◦ u∈ D(A). As in the subsection on nonhomogeneous problems, a substantial relaxation of the requirements can be achieved if A generates an analytic semigroup. A crucial role is played here by the domains of fractional powers of generators. Let us denote Xα = D((−A)α ) with usual graph norm. We have the following theorem. Theorem 20. Let U ⊆ R × Xα , 0 < α < 1, be an open set and f : U → X satisfy: ∀(t,x)∈U ∃(t,x)∈V ⊂U,L>0,0<θ≤1 ∀(ti ,xi )∈V ∥f (t1 , x1 ) − f (t2 , x2 )∥ ≤ L(|t1 − t2 |θ + ∥x1 − x2 ∥Xα ), and let an invertible A, satisfying 0 ∈ ρ(A) be the generator of a bounded ◦ ◦ analytic semigroup. If (0, u) ∈ U , then there is t1 = t1 (u) such that (92) has a unique local classical solution u ∈ C([0, t1 [, X) ∩ C 1 (]0, t1 [, X). Moreover, the solution continuously depends on the initial data and is not global in time if either reaches the boundary U or its Xα norm blows up in finite time. In fact, we have a better regularity result. If the constants θ and L are uniform in U , then u ∈ C 1+ν (]0, t1 [, X); that is, ∂t u is Hölder continuous on ]0, t1 [ with ν = min{θ, β} with 0 < β < 1 − α. We formulated the above theorem in the form usually found in the literature on dynamical systems. However, as was pointed out earlier, using Xα = D((−A)α ) often is inconvenient as it may depend on the particular form of A and there is no explicit method to evaluate the norm. However, in [27, Chapter 7] we can find a parallel theory leading to the analogous theorem in which Xα can be any interpolation space discussed in the section on analytic semigroups. In particular, under assumptions of Theorem 20, in any interpolation space Xα , the solution u is a strict solution; that is u ∈ u ∈ C([0, tmax [, D(A)) ∩ ◦ ◦ ◦ C 1 ([0, t1 [, X), provided u∈ D(A) and A u +f (0, u). The last condition follows from the fact that ∂t u, if it exists, is a mild solution of the differentiated equation (93): 36 Jacek Banasiak ◦ ∫t ◦ AGA (t − s)ϕ(s)ds ∂t u(t) = GA (t)A(u +f (0, u)) + (94) 0 where ϕ(s) = f (s, u(s)) is Hölder continuous by the regularization property mentioned above. Then continuity of ∂t u follows from Theorem 8. 3 Transport on graphs Let us recall that we consider a system of equations ∂ u (x, t) = cj ∂x uj (x, t), x ∈ (0, 1), t j uj (x, 0) = fj (x), m ∑ − ϕ+ ϕij ξj cj uj (1, t) = wij ik (γk ck uk (0, t)), t ≥ 0, (95) k=1 − (ϕ− ij )1≤i≤n,1≤j≤m and Φ+ = (ϕ+ where Φ = ij )1≤i≤n,1≤j≤m are, respectively, the outgoing and incoming incidence matrices, while γj > 0 and ξj > 0 are the absorption/amplification coefficients at, respectively, the head and the tail of the edge ej . Theorem 21. The following conditions are equivalent: 1. (A, D(A)) generates a C0 semigroup; 2. Each vertex of G has an outgoing edge. Proof. 1. ⇒ 2. Assume that there is a semigroup (TA (t))t≥0 generated by A and consider a classical solution u(t) = TA (t)f with f ∈ D(A). Suppose that a vertex, say, vi has no outgoing edge. Then, by (11), 0= m ∑ ϕ+ ik γk ck uk (0, t), t > 0. k=1 In particular, uk (x, t) = fk (x + ck t) for 0 ≤ x + ck t ≤ 1 so uk (0, t) = f (ck t) for 0 ≤ t ≤ c1k . Thus 0= m ∑ ϕ+ ik γk ck fk (ck t), 0 ≤ t ≤ c−1 := min{c−1 k }. k=1 There is a sequence (f r )r∈N , f r ∈ D(A), approximating 1 = (1, 1, . . . , 1) in X. Then −1 0 ≤ ∥1|(0,c−1 ) − (fkr (ck ·))1≤k≤m ∥X c∫ m ∑ 1 = ck k=1 ≤ m ∑ k=1 1 ck ∫1 |1 − fkr (z)|dz → 0 0 0 ck |1 − fkr (z)|dz Kinetic models in natural sciences 37 as r → ∞. Since X−convergence implies convergence almost everywhere of a subsequence, we have m ∑ 0= ϕ+ ik γk ck k=1 −1 almost everywhere on (0, c ), and thus everywhere. Since the graph is connected and we assumed that there is no outgoing edge at vi , there must be an incoming edge and thus at least one term of the sum is positive while all other terms are nonnegative. Thus, if there is a vertex with no outgoing edge, then the set of initial conditions satisfying the boundary conditions is not dense in X and thus (A, D(A)) cannot generate a C0 -semigroup. 1. ⇒ 2. It can be proved that, under assumption 2., the boundary conditions can be incorporated into the domain of the operator in the following compact form: D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BGCu(0)}, (96) where B is the adjacency matrix defined in (9). Clearly, (C0∞ (]0, 1[))m ⊂ D(A) and hence D(A) is dense in X. Let us consider the resolvent equation for A. We have to solve λuj − cj ∂x uj = fj , j = 1, . . . , m, x ∈ (0, 1), with u ∈ D(A). Integrating, we find the general solution cj uj (x) = cj e λ cj x ∫1 λ e cj vj + (x−s) fj (s)ds, (97) x ( λ ) s where v = (v1 , . . . , vm ) is an arbitrary vector. Let Eλ (s) = diag e cj . 1≤j≤m Then (97) takes the form ∫1 Cu(x) = CEλ (x)v + Eλ (x − s)f (s)ds. x To determine v such that u ∈ D(A), we use the boundary conditions. At x = 1 we obtain Cu(1) = CEλ (1)v and at x = 0 ∫1 Cu(0) = Cv + Eλ (−s)f (s)ds 0 so that 38 Jacek Banasiak KCEλ (1)v = KCu(1) = BGCu(0) = BG Cv + ∫1 Eλ (−s)f (s)ds , 0 which can be written as (I − Eλ (−1)C −1 −1 K BGC)v = Eλ (−1)C −1 −1 K ∫1 BG Eλ (−s)f (s)ds. 0 Since the norm of Eλ (−1) can be made as small as one wishes by taking large λ, we see that v is uniquely defined by the Neumann series provided λ is sufficiently large and hence the resolvent of A exists. We need to find an estimate for it. First we observe that the Neumann series expansion ensures that A is a resolvent positive operator and hence the norm estimates can be obtained using only nonnegative entries. Next, we recall that B is column stochastic; that is, each column sums to 1. Adding together the rows in ∫1 KCEλ (1)v = BGCv + BG Eλ (−s)f (s)ds. 0 we obtain m ∑ λ cj ξj cj e vj = j=1 m ∑ γj cj vj + j=1 m ∑ ∫1 γj j=1 − cλ s e j fj (s)ds. 0 Integrating (97) for j ∈ {1, . . . , m}, we obtain ∫1 ∫1 uj (x)dx = vj 0 e λ cj x 1 dx + cj ∫1 ∫1 0 0 λ e cj (x−s) fj (s)ds x ) 1 ∫1 ( ) vj cj ( cλj −λs = e −1 + 1 − e cj fj (s)ds λ λ 0 so that, introducing a weighted space XΞ with the norm ∥u∥Ξ = m ∑ j=1 we have ξj ∥uj ∥L1 ([0,1]) , Kinetic models in natural sciences ∥u∥Ξ = m ∑ 39 ∫1 ξj j=1 uj (x)dx (98) 0 ∫1 ( m m ( λ ) 1∑ ) 1∑ −λs cj ξj vj cj e − 1 + ξj 1 − e cj fj (s)ds = λ j=1 λ j=1 0 1 = λ m ∑ 1 cj vj (γj − ξj ) + λ j=1 m ∑ j=1 ∫1 (γj − ξj ) − cλ s e 0 j 1∑ fj (s)ds + ξj λ j=1 m ∫1 fj (s)ds. 0 We consider three cases (the first one essentially coincides with [18, Proposition 3.3]). (a) γj ≤ ξj for j = 1, . . . , m. Let us consider the iterates in the Neumann series for v, (Eλ (−1)C−1 K−1 BGC)n . Using the fact that C, G and K are diagonal so that they commute, we find Eλ (−1)C−1 K−1 BGC ≤ (CK)−1 Eλ (−1)B(CK). Since B is (column) stochastic, we have r(Eλ (−1)C−1 K−1 BGC) < 1 for any λ > 0. Hence R(λ, A) is defined and positive for any λ > 0. Under the assumption of this item, by dropping two first terms in the second line, (98) gives ∫1 m 1∑ 1 ∥u∥Ξ ≤ ξj fj (s)ds = ∥f ∥Ξ , λ > 0. λ j=1 λ 0 Since D(A) is dense in X, (A, D(A)) generates a positive semigroup of contractions in X. (b) γj ≥ ξj for j = 1, . . . , m. Then (98) implies that for some λ > 0 and c = 1/λ we have ∥R(λ, A)f ∥Ξ ≥ c∥f ∥Ξ and, by density of D(A), the application of the Arendt-Batty-Robinson theorem, Theorem 10, gives the existence of a positive semigroup generated by A in XΞ . Since, however, the norm ∥ · ∥Ξ and the standard norm ∥ · ∥ are equivalent, we see that A generates a positive semigroup in X. (c) γj < ξj for j ∈ I1 and γj ≥ ξj for j ∈ I2 , where I1 ∩ I2 = ∅ and I1 ∪ I2 = {1, . . . , m}. Let L = diag(lj ) where lj = ξj for j ∈ I1 and lj = γj for j ∈ I2 . Then Eλ (−1)C−1 K−1 BGC ≤ (CK)−1 Eλ (−1)B(CL). Thus, if we denote by AL the operator given by the expression A restricted to D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BLCu(0)}, we see that 40 Jacek Banasiak 0 ≤ R(λ, A) ≤ R(λ, AL ) (99) for any λ for which R(λ, AL ) exists. But, by item (b), AL generates a positive semigroup and thus satisfies the Hille–Yosida estimates. Since clearly (99) yields Rk (λ, A) ≤ Rk (λ, AL ) for any k ∈ N, for some ω > 0 and M ≥ 1 we have ∥Rk (λ, A)∥ ≤ ∥Rk (λ, AL )∥ ≤ M (λ − ω)−k , λ>ω and hence we obtain the generation of a semigroup by A. Thus the proof of the theorem is complete. ⊓ ⊔ 4 Epidemiology In this chapter, we simplify the SIRS model given by (17) and consider the SIS model describing the evolution of epidemics which does not convey any immunity. Setting γ = 0 in the system (17) and thus discarding the ‘recovered’ class, we have ∂t s(a, t) + ∂a s(a, t) = −µ(a)s(a, t) − Λ(a, i(·, t))s(a, t) + δ(a)i(a, t), ∂t i(a, t) + ∂a i(a, t) = −µ(a)i(a, t) + Λ(a, i(·, t))s(a, t) − δ(a)i(a, t), ∫ ω s(0, t) = β(a) {s(a, t) + (1 − q)i(a, t)} da, 0 ∫ ω i(0, t) = q β(a)i(a, t) da, (100a) (100b) (100c) (100d) 0 s(a, 0) = s0 (a) = ϕs (a), (100e) i i(a, 0) = i0 (a) = ϕ (a), (100f) for 0 ≤ t ≤ T ≤ +∞, 0 ≤ a ≤ ω ≤ +∞. The force of infection is defined by (18) or (19); the concrete assumption will be introduced when needed. In both cases we deal with a semilinear problem; that is, with a nonlinear (algebraic) perturbation of a linear problem. As in Section 2.7, the decisive role is played by the semigroup generated by the linear part of the problem. Problems like (100a)-(100f) have been relatively well-researched, including the cases where µ and β are nonlinear functions depending on the total population, see [17, 31] and reference therein. Our model most resembles that discussed in [31], the main difference being that in op. cit. the maximum age ω is infinite, thus µ is bounded. However, a biologically realistic assumption is that ω < +∞. This, however, necessitates building into the model a mechanism ensuring that no individual can live beyond ω. It follows, e.g. [21], that the probability of survival of an individual till age a is given by Π(a) = e− Thus Π(ω) = 0 which requires ∫a 0 µ(s) ds . Kinetic models in natural sciences ∫ 41 ω µ(s) ds = +∞. (101) 0 Hence, µ cannot be bounded as a → ω − . This is in contrast with the case ω = +∞, where commonly it is assumed that µ is a bounded function on R+ , and introduces another unbounded operator in the problem. We note that this difficulty was circumvented by Inaba in [22] by introducing the maximum reproduction age a† < ω and ignoring the evolution of the post-reproductive part of the population. Also, in papers such as [17], though ω < +∞, the assumption that the population is constant removes the death coefficient from the equation. The analysis of the model without any simplification in the scalar and linear case was done in [21] by reducing it to an integral equation along characteristics. It can be proved that the solution of such a problem is given by a strongly continuous semigroup. Here we shall prove this directly by refining the argument of [22]. Notation and assumption We will work in the space X = (L1 ([0, ω]))2 with norm ∥(p1 , p2 )∥X = ∥p1 ∥ + ∥p2 ∥, where the norm ∥·∥ refers to the norm in L1 ([0, ω]); the relevant norm in R2 will be denoted by | · |. We also introduce necessary assumptions (cf. [21]) on the coefficients of (100). ( ) (H1) 0 ≤ µ ∈ L∞,loc [0, ω) , satisfying (101), with 0 < µ ≤ µ(a); ( ) (H2) 0 ≤ β ∈ L∞ [0, ω] with β(a) ≤ β := esssup β(a); a∈[0,ω] ( ) (H3) 0 ≤ δ ∈ L∞ ([0, ω] with δ ≤ δ(a) ≤ δ; ) (H4) 0 ≤ K ∈ L∞ [0, ω]2 . ( ) functions. FurLet (W11 [0, ω] )2 be the Sobolev space of vector valued ( ) ther, we define S = diag {−∂a , −∂a } on D(S) = (W11 [0, ω] )2 , Mµ (a) = diag {−µ(a), −µ(a)} on D(Mµ ) = {φ ∈ X : µφ ∈ X}, ( ) 0 δ(a) Mδ (a) = ; (102) 0 −δ(a) Mδ ∈ L(X). Further ( B(a) = β(a) (1 − q)β(a) 0 qβ(a) ∫ with ) (103) ω Bφ = B(a)φ(a) da; 0 the operator B is bounded. Moreover, we introduce the linear operator Aµ defined on the domain 42 Jacek Banasiak D (Aµ ) = {φ ∈ D(S) ∩ D(Mµ ); φ(0) = Bφ} (104) Aµ = S + Mµ . (105) by Let Q be the linear operator defined on the domain D(Q) = D(Aµ ) by Q = Aµ + Mδ . Using these notations, we re-write (100a)-(100f) in the following compact form ∂t u = Aµ u + Mδ u + F(u), ∫ ω B(a)u(a, t) da, u(0, t) = (106a) (106b) 0 u(a, 0) = u0 (a) = φ(a), (106c) where u = (s, i)T and F is a nonlinear function defined by ( ) ( ) −Λ(u) 0 F u = , Λ(u) 0 4.1 The linear part To prove that (100a)-(100f) is well-posed in X, first we show that the linear operator Q on D(Q) = D(Aµ ) generates a strongly continuous positive semigroup on X. Theorem 4.1. The linear operator Q generates a strongly continuous positive semigroup (TQ (t))t≥0 in X. To carry out the proof of Theorem 4.1, it is sufficient to prove the generation result for Aµ and use Theorem 12 (the bounded perturbation theorem) to prove the generation for Q; then we use some other tools to show the positivity of the combined semigroup In this setting, the following result holds. Lemma 4.2. operator Aµ generates a strongly continuous positive ( The linear ) semigroup TAµ (t) t≥0 in X such that ∥T Aµ (t)∥L(X) ≤ e(β−µ)t . (107) To prove the lemma we construct and estimate the resolvent of Aµ . First we introduce the survival rate matrix L(a), which represents the survival rate function in a multi-state population. L(a) is a solution of the matrix differential equation: dL (a) = Mµ (a)L(a), da L(0) = I, (108) Kinetic models in natural sciences 43 where I denotes the 2 × 2 identity matrix. The solution of (108) is a diagonal matrix given by ∫a L(a) = e− 0 µ(r) dr I. (109) We see that ∫a L−1 (a) = e 0 µ(r) dr I. (110) Hence, we can define the resolvent (or state transition matrix, or principal fundamental matrix) L(a, b) as L(a, b) = L(a)L−1 (b). −1 Lemma 4.3. If λ > β − µ, then (λI − Aµ ) is given by −1 φ = (λI − Aµ ) ψ ( )−1 ∫ ω ∫ ω ∫ a e−λa B(a)L(a) = e−λa L(a) I − e−λσ B(σ)L(σ) dσ eλσ L−1 (σ)ψ(σ) dσ da 0 0 0 ∫ a + e−λa L(a) eλσ L−1 (σ)ψ(σ) dσ. (111) 0 Proof. Let λ > β − µ and ψ ∈ X. A function φ ∈ D(Aµ ) if and only if λφ(a) + d φ(a) − Mµ (a)φ(a) = ψ(a), da ∫ (112a) ω φ(0) = B(a)φ(a) da, (112b) 0 µφ ∈ X. By Duhamel’s formula, (112a) leads to ∫ a φ(a) = e−λa L(a)φ(0) + e−λ(a−s) L(a, s)ψ(s) ds, 0 ∫ a ∫ ∫a −λa− 0a µ(r) dr =e φ(0) + e−λ(a−s)− s µ(r) dr ψ(s) ds, (112c) (113) 0 for some unspecified as yet initial condition φ(0). For a fixed φ(0) we denote φ := Rφ(0) (λ)ψ; it is easy to see that (λI − S − Mµ )Rφ(0) (λ)ψ = ψ, (114) for a.a. a ∈ [0, ω). The unknown φ(0) can be determined from (112b) by substituting (113); we get ∫ ω ∫a φ(0) = e−λa− 0 µ(r) dr B(a)φ(0) da 0 (∫ a ) ∫ ω ∫ ∫ −λa− 0a µ(r) dr λs+ 0s µ(r) dr + e B(a) e ψ(s) ds da. 0 0 44 Jacek Banasiak Since ∫ ω −λa− e ∫a 0 µ(r) dr 0 ∫ B(a) da ≤ β ω e−(λ+µ)a da ≤ 0 β <1 λ+µ (115) ∫a ∫ω for λ > β̄ − µ, I − 0 e−λa− 0 µ(r) dr B(a) da is invertible with ( )−1 ∫ ω ∫ λ+µ −λs− 0s µ(r) dr . e B(s) ds I− ≤ λ − (β − µ) 0 (116) Hence ( )−1 ∫ ω ∫ −λa− 0a µ(r) dr e φ(0) = I − B(a) da 0 (∫ ) ∫ ω a ∫ ∫ −λa− 0a µ(r) dr λs+ 0s µ(r) dr e B(a) e ψ(s) ds da 0 0 and we can substitute that φ(0) in the operator Rφ(0) (λ) to define R(λ)ψ(a) −λa− := e ∫a 0 µ(r) dr ( ∫ I− ω ∫ −λs− 0s µ(r) dr e )−1 ∫ 0 ∫ a × eλs+ ∫s 0 µ(r) dr ∫a ψ(s) ds da + e−λa− 0 ω B(s) ds 0 e−λa− ∫a 0 µ(r) dr B(a) 0 ∫ a µ(r) dr eλs+ ∫s 0 µ(r) dr ψ(s) ds. 0 The above calculations show that λ − Aµ is one-to-one for λ > β − µ. Routine calculations show that R(λ)ψ ≤ X 1 ψ . X λ − (β − µ) Thus R(λ) is a bounded operator in X. Further, we also show that ( ∫ ω |µ(a)R(λ)ψ(a)| da ≤ 0 β 1+ λ − (β − µ) (117) ) ψ . X Hence R(λ)X ⊂ D(Mµ ). Further, since for any ψ ∈ X, φ = R(λ)ψ satisfies λR(λ)ψ + d R(λ)ψ − Mµ R(λ)ψ = ψ da almost everywhere, we have d R(λ)ψ = ψ − λR(λ)ψ + Mµ R(λ)ψ, da Kinetic models in natural sciences 45 where, by the above (estimates, all terms on the right hand side are in X. Hence R(λ)ψ ∈ (W11 [0, ω]))2 . Since the boundary condition holds, using the results above, we see that R(λ) : X → D(Aµ ). Then R(λ) is a right-inverse of the operator λ−A, hence R(λ) is the right inverse of (λI − Aµ , D(Aµ )). To prove that it is also a left inverse, we repeat the standard argument. Assume that for some φ ∈ D(Aµ ) we have e ̸= φ. R(λ)(λI − Aµ )φ = φ Since R(λ) : X → D(Aµ ), we can write (λI − Aµ )φ = (λI − Aµ )R(λ)(λ − Aµ )φ = (λ − Aµ )e φ since R(λ) is a right inverse of λI −Aµ . But we proved that the linear operator e and hence R(λ) = (λ − Aµ )−1 . (λ − Aµ ) is one-to-one for λ > β − µ, φ = φ Lemma 4.4. D(A)+ = X+ . Proof. A proof of this result (with gaps) is provided in [22, p. 60]. A more comprehensive proof can be found in [31]. We present a much simpler proof which, moreover, allows for an approximation of ψ ∈ X+ by elements of D(Aµ )+ . Fix f ∈ X+ . First we note that for any given ϵ there is 0 ≤ ϕ ∈ (C0∞ (]0, ω[))2 such that ∥f − ϕ∥X ≤ ϵ. Clearly, ϕ ∈ D(Mµ ) but typically ∫ ω φ(0) ̸= B(a)φ(a) da. 0 C0∞ ([0, ω)) with η(0) = 1 and let ηϵ (a) = η(a/ϵ). Take a function 0 ≤ η ∈ Further, let α = (α1 , α2 ) be a vector and consider ψ = φ + ηϵ α. Clearly, ψ ∈ (W11 ([0, ω]))2 ∩ D(Mµ ). As far as the boundary condition is concerned we have, by the properties of the involved functions, ϵω ) ∫ ω ∫ ( β(a)ηϵ (a) (1 − q)β(a)ηϵ (a) α= B(a)φ(a) da + da α. (118) 0 qβ(a)ηϵ (a) 0 0 Now, since ∫ϵω 0≤ ∫ω the matrix l1 −norm satisfies ϵω ∫ ( β(a)ηϵ (a) 0 0 β(ϵs)η(s)ds ≤ ϵβ̄, β(a)ηϵ (a)da = ϵ 0 0 (1 − q)β(a)ηϵ (a) qβ(a)ηϵ (a) ) da ≤ ϵβ̄. L(R2 ) 46 Jacek Banasiak Thus, (118) is solvable for sufficiently small ϵ, by positivity of the above matrix and the Neumann series representation, α is nonnegative and ∫ ω |α| ≤ B(a)φ(a) da (1 − ϵβ̄)−1 ≤ C 0 for some constant C, which is independent of ϵ for sufficiently small ϵ (∥φ∥, which depends on ϵ, can be bounded by e.g. ∥f ∥ + 1 for ϵ < 1). Hence we have ∥f − ψ∥X ≤ ∥f − φ∥X + |α|∥ηϵ ∥ ≤ (1 + C)ϵ. Proof of Lemma 4.2. Since the inverse of a bounded operator is closed, we see that λI − Aµ , and hence Aµ , are closed. Thus the above lemmas with the estimate (117) show that Aµ satisfies the assumptions of the Hille–Yosida theory. Hence, it generates a semigroup satisfying (107). Since the resolvent is positive, the semigroup is positive as well. ⊓ ⊔ Proof of Theorem 4.1. Since Mδ ∈ L (X), with Mδ (a)L(X) ≤ 2δ, Theorem 12 (the bounded perturbation theorem) is applicable and states that the linear (operator) (Q, D(A)) generates a strongly continuous semigroup denoted by T Q (t) t≥0 . Using the estimate (107) we have: ∥T Q (t)∥L(X) ≤ et(β−µ+2δ̄) . Using the structure of Mδ we can improve this estimate and also show that the semigroup (T Q (t))t≥0 is positive. Since the the variable a plays in Mδ the role of a parameter, we find ( ) 1 1 − e−tδ(a) T Mδ (t) = , 0 e−tδ(a) and so we have ∥T Mδ (t)∥L(X) = 1. Also, (T Mδ (t))t≥0 is positive. Hence, by (76), we obtain ∥T Q (t)∥L(X) ≤ et(β−µ) (119) ⊓ ⊔ and (T Q (t))t≥0 is positive. Remark 6. The estimates (107) and (119) are not optimal. In fact, for the scalar problem ∂t n(a, t) = −∂a n(a, t) − µ(a)n(a, t), ∫ω n(0, t) = β(a)n(a, t)da, 0 ◦ n(a, 0) = n (a), t > 0, a ∈]0, ω[ (120) Kinetic models in natural sciences 47 it can be proved, [21], that there is a unique dominant eigenvalue λ∗ of the problem such that ∗ ∥n(t)∥ ≤ M etλ , (121) which is the solution of the renewal equation ∫ω 1= ∫a −λa− µ(s)ds β(a)e 0 da. (122) 0 This eigenvalue is, respectively, positive, zero or negative if and only if the basic reproduction number ∫ω R= − β(a)e ∫a 0 µ(s)ds da (123) 0 is bigger, equal, or smaller, than 1. ◦ ◦ Consider now an initial condition (s, i ) ∈ D(Aµ )+ . Since the semigroup (T Q (t))t≥0 is positive, the strict solution (s, i) of the linear part of (100) is nonnegative and the total population 0 ≤ s(a, t) + i(a, t) = n(a, t) satisfies (120). Using nonnegativity, we find s(a, t) ≤ n(a, t) and i(a, t) ≤ n(a, t) and consequently ∗ ◦ ◦ ◦ ◦ ∥T Q (t)( s, i )∥X ≤ M etλ ∥(s, i )∥X ◦ ◦ for (s, i ) ∈ D(Aµ )+ . However, by Lemma 4.4, the above estimate can be extended to X+ and, by (29), to X. Note that the crucial role in the above argument is played by the fact that (s, i) satisfies the differential equation (100) — if it was only a mild solution, it would be difficult to directly prove that the sum s + i is the mild solution to (120). 4.2 The nonlinear problem Intercohort Transmission In the case of intercohort transmission, individuals of any age can infect individuals of any age, though with possibly different intensity. Then ∫ ω Λ(a, i) = K(a, a′ )i(a′ ) da′ , (124) 0 ′ where K(a, a ) is a nonnegative bounded function on [0, ω] × [0, ω] which account for the age-specific (average) probability of becoming infected through contact with infectives of a particular age. Since the nonlinear term F is quadratic, it is easy to see that the following result is true. 48 Jacek Banasiak Proposition 4.5. F is( continuously ) (Fréchet) differentiable with respect to ϕ ∈ X and for any ϕ = ϕs , ϕi , ψ = ψ s , ψ i ∈ X the Fréchet derivative at ϕ, Fϕ , is defined by ∫ω ∫ω s ′ i ′ ′ s ′ i ′ ′ −ψ (a) K (a, a ) ϕ (a ) da − ϕ (a) K (a, a ) ψ (a ) da ( ) 0 0 . Fϕ ψ (a) := ω ∫ω ∫ s ′ i ′ ′ s ′ i ′ ′ ψ (a) K (a, a ) ϕ (a ) da + ϕ (a) K (a, a ) ψ (a ) da 0 0 (125) ◦ ◦ ◦ This allows for applying Theorem 19 to claim that for each u = (s, i ) ∈ X, ◦ there is a t(u) such that the problem (106) has a unique mild solution on ◦ ◦ [0, t(u)[∋ t → u(t); this solution is a classical solution if u ∈ D(Aµ ). We recall that that the proof consists of showing that the Picard iterates ◦ u0 = u ◦ ∫t un (t) = T Q (t) u + T Q (t − s)F(un−1 (s))ds (126) 0 ◦ ◦ ◦ ◦ converge in C([0, t(u)[, B(u, ρ)) where B(u, ρ) = {u ∈ X; ∥u− u ∥X ≤ ρ} for some constant ρ. Since the nonlinearity is quadratic, it is not globally Lipschitz continuous and thus the question whether this solution can be extended to [0, ∞[ requires employing positivity techniques. Since F is not positive on X+ , we cannot claim that the constructed local solution is nonnegative, as the iterates defined by (126) need not to be positive, ◦ even if we start with u≥ 0. Hence, we re-write (106) in the following equivalent form ( ) du = (Q − κI) u + (κI + F) u , t > 0, (127) dt ◦ u(0) = u, for some κ ∈ R+ to be determined. Denote Qκ = Q−κI; then (T Q,κ (t))t≥0 = (e−κt T Q )t≥0 and hence (T Q,κ (t))t≥0 is positive. Thus the mild solution satisfies ∫ t ( ) ◦ ◦ u(t) = e−κt T (t) u + e−κ(t−s) TQ (t − s) κI + F (u(s)) ds, 0 ≤ t < t(u). 0 (128) It is easy to see that the following result holds. ( ) ◦ Lemma 4.6. For any ρ there exists κ such that κI+F (X+ ∩B(u, ρ)) ⊂ X+ . With this result, the Picard iterates for (128) are nonnegative and we can repeat the standard estimates to arrive at the following result: Kinetic models in natural sciences ◦ 49 ◦ Theorem 4.7. Assume that u∈ X+ and let u : [0, t(u)[→ X be the unique mild solution of (106). Then this solution is nonnegative on the maximal interval of its existence. Moreover, the solutions continuously depend on the initial conditions on every compact time interval of their joint interval of existence. 4.3 Global existence Since quadratic nonlinearities do not satisfy the uniform Lipschitz condition, we cannot immediately claim that the solutions to (106) are global in time. In fact, it is well known that, even for ordinary differential equations, the solution with a quadratic nonlinearity can blow in finite time. Here, we use positivity to show thatpositive solutions exist globally in time. For this, we have to show that t → u(t)X does not blow up in finite time. We state the following result: ◦ Theorem 4.8. For any u∈ D(Aµ ) ∩ X+ , the problem (106) has a unique strict positive solution u(t) defined on the whole time interval [0, ∞[. Proof. The proof is follows the same idea as in Remark 6. Under( the assump) tions, we have locally defined positive strict solution u(t) = s(t), i(t) to (106) and hence of (100) in (L1 ([0, ω]))2 . Thus ∫ ω ∫ ω u(t) = (s(a, t) + i(a, t)) da = u(a, t) da, X 0 0 where u(a, t) is the solution to the McKendrick equation (120). But then, as ◦ long as for 0 ≤ t < t(u), ◦ u(t) ≤ e(β−µ)t u . X X Accordingly, u(t)X does not blow up in finite time and hence the solution is global. ◦ Corollary 2. For any u∈ X+ , the problem (106) has a unique mild positive solution u(t) defined on the whole time interval [0, ∞[. The proof follows from D(Aµ )+ = X+ and the continuous dependence on initial conditions. 4.4 Intracohort Transmission In this section, we are concerned with the situation where the disease transmission interactions are restricted to individuals of the same range of age. A constitutive form of the infection rate relative to this mechanism of transmission is provided in [16, p. 1381]: Λ(a, i(·, t)) = K0 (a)i(a, t). (129) 50 Jacek Banasiak Therefore, the nonlinear term (also known as forcing term) F in (106) is defined by [ ] ( ) −K0 (a)s(a, t)i(a, t) F u(a, t) = , u = (s, i). K0 (a)s(a, t)i(a, t) In the sequel, we use the notation below. X1 = L1 ([0, ω], R2 ), X∞ = L∞ ([0, ω], R2 ), ( ) Y∞ = C [0, T ], X∞ , The main problem with the intracohort transmission is that, in general, F(u) ∈ / X1 for u ∈ X1 . Multiplication is well defined in L∞ ([0, ω]) but then the latter space is not suitable for semigroup techniques — any C0 -semigroup on L∞ is uniformly continuous. To handle this nonlinearity, we use the fact that for ω < ∞, X∞ is a subspace densely and continuously embedded in X1 and show that we can restrict the analysis performed in the previous section to X∞ . First we note the following result. ◦ Proposition 4.9. For any t ≥ 0, TQ (t) (X∞ ) ⊂ X∞ with ∥TQ (t) u∥X∞ ≤ ◦ ◦ βωeβω ∥u∥X∞ for all u∈ X∞ . This result follows from the integral representation of the semigroup. ) Now, let u(t) = (s(·, t), i(·, t) be a function in X∞ . Under the above setting, (100a)-(100f) is rewritten, as in the previous section, in the following abstract differential equation in a Banach space X∞ : du = Qu + F (u) , t > 0, (130) dt ◦ u(0) =u . We observe that the nonlinear function F has the following properties: ( ) • F(X∞ ) ⊂ X∞ with ∥F u ∥X∞ ≤ ∥K0 ∥∞ ∥u∥2X∞ ; • F is continuously Fréchet differentiable with respect to φ ∈ X∞ hence, in particular, it is locally Lipschitz continuous. Using this properties and Proposition 4.9 we can show that Picard iterates of the integral formulation of (130): ∫ t ( ) ◦ (u) (t) = TQ (t) u + TQ (t − s)F u(s) ds, 0 < t < T, (131) 0 stay in an an appropriately chosen closed ball in Y∞ for sufficiently small T > 0 and converge to a unique fixed point which is thus a mild solution of (130). Kinetic models in natural sciences 51 Restricting considerations to such a ball, we can consider the problem with the linear L1 semigroup to justify calculations and, in the same way as in the intercohort case, prove positivity, global existence and classical solvability for ◦ u∈ D(Aµ ) ∩ X∞ . 5 Coagulation-fragmentation equation Recall that we deal with the equation ∫∞ ∂t u(x, t) = −a(x)u(x, t) + a(y)b(x|y)u(y, t)dy x ∫∞ −u(x, t) k(x, y)u(y, t)dy 0 + 1 2 ∫x k(x − y, y)u(x − y, t)u(y, t)dy, (132) 0 where x ∈ R+ :=]0, ∞[ is the size of the particles/clusters. Here u is the density of particles of mass/size x, a is the fragmentation rate and b describes the distribution of masses x of particles spawned by fragmentation of a particle of mass y. Thanks to the work of P. Laurençot, [26], there is a well developed theory of weak solvability of coagulation–fragmentation problems. The main aim of this section is to present results on strict solvability of them. Fragmentation rates. The fragmentation rate a is assumed to satisfy 0 ≤ a ∈ L∞,loc ([0, ∞[), (133) that is, we allow a to be unbounded as x → ∞. Further, b ≥ 0 is a measurable function satisfying b(x|y) = 0 for x > y and (24). The expected number of particles resulting from a fragmentation of a size y parent, denoted by n0 (y), is assumed to satisfy n0 (y) < +∞ (134) for any fixed y ∈ R+ . We impose some control on the growth of the fragmentation rates. Namely, we assume that there are j ∈]0, ∞[, l ∈ [0, ∞[ and a0 , b0 ∈ R+ such that for any x ∈ R+ a(x) ≤ a0 (1 + xj ), n0 (x) ≤ b0 (1 + xl ). (135) 52 Jacek Banasiak Coagulation rates. The coagulation kernel k(x, y) represents the likelihood of a particle of size x attaching itself to a particle of size y. We assume that it is a measurable symmetric function such that for some K > 0 and 0 ≤ α < 1 0 ≤ k(x, y) ≤ K((1 + a(x))α (1 + a(y))α . (136) This will suffice to show local in time solvability of (132) whereas to show that the solutions are global in time we need to strengthen (136) to 0 ≤ k(x, y) ≤ K((1 + a(x))α + (1 + a(y))α ) (137) for some 0 ≤ α < 1. State spaces. In fragmentation and coagulation problems, two spaces are most often used due to their physical relevance.∫ In the space L1 (R+ , xdx) the norm of a non∞ negative element u, given by 0 u(x)xdx, represents the total mass of the system, whereas the norm of a nonnegative element u in the space L1 (R+ , dx), ∫∞ u(x)dx, gives the total number of particles in the system. 0 We use the scale of spaces with finite higher moments Xm = L1 (R+ , dx) ∩ L1 (R+ , xm dx) = L1 (R+ , (1 + xm )dx), (138) where m ∈ M := [1, ∞[. We extend this definition to X0 = L1 (R+ ). We note that, due to the continuous injection Xm ,→ X1 , m ≥ 1, any solution in Xm is also a solution in the basic space X1 . Thus, we denote by ∥ · ∥m the natural norm in Xm defined in (138). To shorten notation, we define wm (x) := 1 + xm . 5.1 Main Results To formulate these results, we have to introduce specific assumptions and notation. First we define ∫y b(x|y)xm dx nm (y) := 0 for any m ∈ M0 := {0} ∪ M and y ∈ R+ . Further, let N0 (y) := n0 (y) − 1 and Nm (y) := y m − nm (y), m ≥ 1. Kinetic models in natural sciences 53 It follows that N0 (y) = n0 (y) − 1 ≥ 0 and ∫y Nm (y) = y m ∫y − b(x|y)x dx ≥ y m m −y m−1 0 b(x|y)xdx = 0 (139) 0 for m ≥ 1 with N1 = 0. The fragmentation part Next, for any m ∈ M, let (Am u)(x) := a(x)u(x) on D(Am ) = {u ∈ Xm : au ∈ Xm } and let Bm be the restriction to D(Am ) of the integral expression ∫∞ [Bu](x) = a(y)b(x|y)u(y)dy. x Theorem 22. Let a, b satisfy (24), (134) and (135), and let m be such that m ≥ j + l if j + l > 1 and m > 1 if j + l ≤ 1. a) The closure (Fm , D(Fm )) = (−Am + Bm , D(Am )) generates a positive quasi-contractive semigroup, say (SFm (t))t≥0 , of type at most 4a0 b0 on Xm . Furthermore, if u ∈ D(Fm )+ , then Nm (x)a(x)u(x) ∈ X0 , m ∈ M0 . (140) b) If, moreover, for some m there is cm > 0 such that lim inf x→∞ Nm (x) = cm , xm (141) then Fm = −Am + Bm and (SFm (t))t≥0 is an analytic semigroup on Xm . c) If (141) holds for some m0 , then it holds for all m ≥ m0 . We note that (141) cannot hold for m = 1 as N1 = 0. Proof. We shall fix m satisfying m ≥ j + l if j + l > 1 and m > 1 otherwise; see (135). First we show that Bm := B|D(Am ) is well defined. Next, direct integration gives for u ∈ D(Am ) ∫∞ (−Am + Bm )u(x)wm (x)dx = −ϕm (u) 0 ∫∞ := (N0 (x) − Nm (x))a(x)u(x)dx, 0 (142) 54 Jacek Banasiak If the term N0 (x) > 0 had not been present, then (142) would have allowed a direct application of the substochastic semigroup theory. In the present case we note that u ∈ D(Am )+ we have, by (139), ∫∞ −ϕm (u) ≤ ∫∞ N0 (y)a(y)u(y)dy ≤ 4a0 b0 0 u(x)wm (x)dx =: η∥u∥m , 0 ∫∞ Then we have ϕem (u) := ϕm (u) + η 0 u(x)wm (x)dx ≥ 0 for 0 ≤ u ∈ D(Am ) em , D(Am )) := (Am + ηI, D(Am )) satisfies and the operator (A ∫∞ em + Bm )u(x)wm (x)dx = −ϕem (u) (−A 0 ∫ ∞ = −η 0 ∫∞ u(x)wm (x)dx + (N0 (x) − Nm (x))a(x)u(x)dx 0 ≤ 0. em +Bm generates a substochastic semigroup (S e (t))t≥0 An extension Fem of −A Fm and thus there is an extension Fm of (−Am +Bm , D(Am )) given by (Fm , D(Fm )) = (Fem +ηI, D(Fem )) generating a positive semigroup (SFm (t))t≥0 = (eηt SFem (t))t≥0 on Xm . Furthermore, ϕem extends to D(Fm ) by monotone limits of elements of D(Am ). Thus, let u ∈ D(Fm )+ with D(Am ) ∋ un ↗ u. Then, since ∫∞ ∫∞ N0 (x)a(x)u(x)dx < ∞, u(x)wm (x)dx < ∞, 0 0 by (135), m ≥ j + l and D(Fm ) ⊂ Xm , and the fact that ϕem (un ) tends to a finite limit, we have ∫∞ lim ∫∞ Nm (x)a(x)un (x)dx = n→∞ 0 Nm (x)a(x)u(x)dx < +∞. 0 To show that (Fm , D(Fm )) = (−Am + Bm , D(Am )) we use Theorem 17. In can be proved, [6, Theorem 6.20], that the generator Fm satisfies Fm ⊂ Fm,max where Fm,max = A + B restricted to D(Fm,max ) = {u ∈ Xm ; Au + Bu ∈ Xm }. Note that neither Au nor Bu are guaranteed to be in Xm . Since au ∈ L1 ([0, R], (1 + xm )dx) for any 0 < R < +∞, therefore the same is true for Bu. Hence, with wm (x) = 1 + xm , Kinetic models in natural sciences 55 ∫∞ (−a(x)u(x) + [Bu](x)) wm (x)dx 0 = lim − ∫R ∫R a(x)u(x)wm (x)dx + R→∞ 0 ∫∞ a(y)b(x|y)u(y)dy wm (x)dx . x 0 Next, by (24), ∫R ∫∞ ∫R ∫y a(y)b(x|y)u(y)dy wm (x)dx = b(x|y)wm (x)dx u(y)a(y)dy x 0 ∫∞ + R ∫R 0 b(x|y)wm (x)xdx u(y)a(y)dy = 0 0 ∫R a(y)u(y)(n0 (y) + nm (y))ydy + SR , 0 where SR ≥ 0. Combining this with (142), we see that ∫∞ (−a(x)u(x) + [Bu](x)) wm (x)dx 0 ∫∞ = (N0 (x) − Nm (x))a(x)u(x)dx + lim SR ≥ −ϕm (u), R→∞ 0 so that Theorem 17 gives the thesis. To prove part b), we begin by observing that inequality (139) implies that 0 ≤ Nm (x) ≤ xm . This, together with (141), yields cm xm /2 ≤ Nm (x) ≤ xm for large x which, by (140), establishes that if u ∈ D(Fm ), then au ∈ Xm or, in other words, that D(Fm ) ⊂ D(Am ). Since (Fm , D(Fm )) is an extension of (−Am + Bm , D(Am )), we see that D(Fm ) = D(Am ). It is clear that the semigroup generated by −Am is bounded. Furthermore, if λ = r + is, then |λ + a(x)|2 ≥ s2 and therefore, for all r > 0 ∥R(r + is, −Am )f ∥m ∫∞ 1 |f (x)|(1 + xm )dx ≤ 1 ∥f ∥m . = r + is + a(x) |s| 0 The analyticity of the fragmentation semigroup then follows from the ArendtRhandi theorem. Example 7. One of the forms of b(x|y) most often used in applications is ( ) 1 x b(x|y) = h (143) y y 56 Jacek Banasiak which is referred to as the homogeneous fragmentation kernel. In this case the distribution of the daughter particles does not depend directly on their relative sizes but on their ratio. In this case ∫y ( ) ∫1 1 x m m nm (y) = h x dx = y h(z)z m dz =: hm y m . y y 0 0 Since 1 y = n1 (y) = y ∫y ( ) ∫1 x h xdx = y h(z)zdz = h1 y y 0 0 we have h1 = 1 so that hm < 1 for any m > 1 and Nm (y) = y m (1 − hm ). Hence, (141) holds. On the other hand, fragmentation processes in which daughter particles tend to accumulate close both to 0 and to the parent’s size may not satisfy (141). The coagulation part Next, we introduce a nonlinear operator Cm in Xm defined for u from a suitable subset of Xm by the formula ∫∞ (Cm u)(x) := −u(x) 1 k(x, y)u(y)dy + 2 0 ∫x k(x − y, y)u(x − y)u(y)dy 0 so that the initial value problem for (132) can be written as an abstract semilinear Cauchy problem in Xm ∂t u = −Am u + Bm u + Cm u, ◦ u(0) = u . (144) To formulate the main theorems we have to introduce a new class of spaces which, as we shall see later, is related to intermediate spaces associated with the fragmentation operator Fm and its fractional powers. We set { } ∫ ∞ (α) α m Xm := u ∈ Xm ; |u(x)|(ω + a(x)) (1 + x ) dx < ∞ , (145) 0 where ω is a sufficiently large constant. Then we have Theorem 23. Assume that a, b, k satisfy (24), (134), (135), (136)and (141) ◦ (α) for some m0 > 1, and let m ≥ max{j + l, m0 } hold. Then, for each u ∈ Xm,+ , there is τ > 0 such that the initial value problem (144) has a unique nonnega(α) tive classical solution u ∈ C([0, τ ], Xm ) ∩ C 1 ((0, τ ), Xm ) ∩ C((0, τ ), D(Am )). Furthermore, there is a measurable representation of u which is absolutely continuous in t ∈ (0, τ ) for any x ∈ R+ and which satisfies (132) almost everywhere on R+ × (0, τ ). Kinetic models in natural sciences 57 Finally, for global in time solvability we need to restrict the growth rate of k. Namely, we have Theorem 24. Let the assumptions of Theorem 23 hold with β = 0, that is, let k satisfy (137). Furthermore, let the constant j from assumption (135) be such that αj ≤ 1. Then any local solution of Theorem 23 is global in time. Interlude – intermediate spaces associated with Fm . Define Fm,ω := Fm − ωI, Am,ω := Am + ωI, D(Fm,ω ) = D(Fm ) = D(Am ) = D(Am,ω ), (146) where ω > 4a0 b0 is a fixed constant. The operators (Fm,ω , D(Am )) and (−Am,ω , D(Am )) generate analytic semigroups (SFm,ω (t))t≥0 = (e−ωt SFm (t))t≥0 and (S−Am,ω (t))t≥0 = (e−ωt S−Am (t))t≥0 on Xm . Since each operator is invertible, the norms ∥u∥m,A := ∥Am,ω u∥m and ∥u∥m,F := ∥Fm,ω u∥m , u ∈ D(Am ) are equivalent to each other and also to the corresponding graph norms on D(Am ). Then we have (up to the equivalence of the respective norms) DFm,ω (α, r) = D−Am,ω (α, r). (147) We find it most convenient to use D−Am,ω (α, 1) which equals the real interpolation space (Xm , D(Am,ω ))α,1 . It follows that (α) (Xm , D(Am,ω ))α,1 =: Xm { } ∫ ∞ α m = u ∈ Xm ; ∥u∥(α) = |u(x)|(ω + a(x)) (1 + x ) dx < ∞ , m 0 Indeed, Let u ∈ Xm . Then, by the Fubini-Tonelli theorem and routine changes of variables, we obtain ω+a(x) ∫ ∞ ∫ ∥t−α Am,ω S−Am,ω (t)u∥L1 (J) = |u(x)|(ω+a(x))α s−α e−s ds wm (x)dx 0 0 with ∫ω 0< −α −s s 0 e ω+a(x) ∫ ds ≤ −α −s s e ∫∞ ds ≤ 0 s−α e−s ds = Γ (1 − α). 0 In other words, there is a constant c1 ≥ 1 such that (α) (α) c−1 1 ∥u∥m ≤ ∥u∥DFm,ω (α,1) ≤ c1 ∥u∥m , ∀ u ∈ DFm,ω (α, 1). (148) 58 Jacek Banasiak Proof of local solvability Here we assume that a and b satisfy the assumptions of Theorem 22 b) so that, in particular, (141) holds for some m ≥ j + l or m > 1 if j + l ≤ 1. Furthermore, the coagulation kernel is such that (136) is satisfied. We fix α ω > max{4a0 b0 , 1} and denote aα ω (x) := (ω + a(x)) . We consider the following modified version of (132) ∫∞ ∂t u(x, t) = −(aω (x) + γaα ω (x))u(x, t) + a(y)b(x|y)u(y, t)dy x ∫∞ +(γaα ω (x) + ω)u(x, t) − u(x, t) k(x, y)u(y, t)dy 0 + 1 2 ∫x k(x − y, y)u(x − y, t)u(y, t)dy, (149) 0 where γ is a constant to be determined and α is the index appearing in (136). Then (Fγ , D(Fγ )) := (Fm,ω − γAα m , D(Am )) generates an analytic semigroup, say (SFγ (t))t≥0 , on Xm . Since (SFm,ω (t))t≥0 and (S−γAαm (t))t≥0 are positive and contractive, we can use the Trotter product formula to deduce that (SFγ (t))t≥0 is also a positive contraction on Xm . Furthermore, since S−γAαm (t) ≤ Id for t ≥ 0, using again the Trotter formula SFγ (t)u ≤ SFm,ω (t)u, u ∈ Xm,+ . (150) (α) and thus, for u ∈ Xm 2 (α) ∥SFγ (t)u∥(α) m ≤ c1 ∥u∥m . (151) Next consider the set (α) U = {u ∈ Xm,+ : ∥u∥(α) m ≤ 1 + b}, (152) for some arbitrary fixed b > 0 and set γ = 2K(b + 1). Then on U we obtain ∫∞ (Cγ u)(x) := −u(x) k(x, y)u(y)dy + (γ(aα ω (x) + ω)u(x) 0 + 1 2 ∫x k(x − y, y)u(x − y)u(y)dy ≥ 0. 0 (153) Kinetic models in natural sciences Similarly, on U we have ∥Cγ u∥m ≤ K1 (U), 59 (154) as well as α (α) ∥(γAα m + ωI)u − (γAm + ωI)v∥m ≤ (ω + γ)∥u − v∥m and ∥Cγ u − Cγ v∥m ≤ K2 (U)∥u − v∥(α) m , ◦ (155) ◦ (α) (α) for some constants K1 (U), K2 (U). Hence, for u ∈ Xm,+ satisfying ∥ u ∥m ≤ ◦ c−2 1 b, for b of (152) and c1 from (151), there is τ = τ (u) such that the mapping method for ∫t ◦ (T u)(t) = SFγ (t) u + SFγ (t − s)Cγ u(s)ds 0 is a contraction on Y = C([0, τ ], U), with U defined by (152) and the metric ◦ (α) (α) induced by the norm ∥u(t)∥Y := sup ∥u(t)∥m . Therefore, for any u ∈ Xm,+ , 0≤t≤τ (α) there is a unique mild solution u to (144) in Xm,+ which, moreover, satisfies u ∈ C 1 ((0, τ ), Xm ) ∩ C((0, τ ), D(Am )). Proof of global solvability. The local solution, constructed in the previous section, can be extended in ◦ a usual way to the maximal forward interval of existence [0, τmax (u)). Thus, (α) to show that u is globally defined, we need to show that ∥u(t)∥m is a priori bounded uniformly in time. Let us denote by Mr the r-th moment of u, ∫∞ xr u(x)dx. Mr (u) := 0 Then, for some constant L, ∫∞ ∥u∥(α) m ≤L |u(x)|(1 + xm+jα )dx = L(M0 (u) + Mm+jα (u)). (156) 0 Though for a given m, Theorem 23 does not ensure the differentiability of Mm+αj , it is valid in the scale of spaces Xr with r ≥ m provided, of course, ◦ (α) u ∈ Xr(α) . Since the embedding Xr(α) ⊂ Xm is continuous for r ≥ m, the 60 Jacek Banasiak ◦ (α) (α) solutions emanating from the same initial value u ∈ Xr ⊂ Xm in each ◦ (α) space, by construction, must coincide. Hence, let u ∈ Xm+jα ⊂ Xm+jα ⊂ (α) Xm so that ◦ ◦ (α) ◦ u ∈ C([0, τmax (u)), Xm+jα )∩C 1 ((0, τmax (u)), Xm+jα )∩C((0, τmax (u)), D(Am+jα )), ◦ with possibly different, but still nonzero, τmax (u). This, in particular, yields differentiability of ∥u(·)∥0 = M0 (u(·)) and, consequently, of Mm+jα (u(·)). To get the moment estimates we use the inequality (x + y)r − xr − y r ≤ (2r − 1)(xr−1 y + y r−1 x) =: Gr (xr−1 y + y r−1 x), (157) for r ≥ 1, x, y ∈ R+ . Then ∫∞ xr (Cu)(x)dx 0 = Gr KLα (Mr+jα−1 M1 + Mr−1 M1+jα + 2Mr−1 M1 ). (158) For the particular cases r = 0 and r = 1 we obtain ∫∞ ∫∞ 1 (Cu)(x)dx = − k(x, y)u(x, t)u(y, t)dxdy ≤ 0, 2 0 0 ∫∞ x(Cu)(x)dx = 0. 0 ◦ Hence, using estimates for the linear part, we obtain on (0, τmax (u)) M0,t ≤ 4a0 b0 (M0 + Mm ), M1,t = 0, (159) Mm+jα,t ≤ Gm+jα KLα (Mm+2jα−1 M1 + Mm+jα−1 (M1+jα + 2M1 )). We see that if 1 ≤ r ≤ r′ , then Mr ≤ M1 + Mr′ (160) ′ as xr ≤ x on [0, 1] and xr ≤ xr on [1, ∞). Thus, we see that in order for the moment system (159) to be closed, we must assume that jα ≤ 1. This allows us to re-write (159) as M0,t ≤ 4a0 b0 (2M0 + Mm+jα ), Mm+jα,t ≤ Gm+jα KLα ((Mm+jα + M1 )M1 +(Mm+jα + M1 )(M2 + 3M1 )), (161) (162) Kinetic models in natural sciences 61 where M1 is constant and where we used jα ≤ 1. To find the behaviour of M2 , again we use (142) and (158), with an obvious simplification of (157), to get the estimate for M2 as M2,t ≤ 4KLα (M1+jα M1 + M12 ) ≤ 4KLα (M2 M1 + 2M12 ). Hence, M2 is bounded on its interval of existence. Then, from the second inequality in (162), we see that Mm+jα satisfies a linear inequality with bounded ◦ coefficients and thus it also is bounded on (0, τmax (u)). This in turn yields the (α) boundedness of M0 . Hence, ∥u(·)∥m is bounded and thus u exists globally. To ascertain global existence of solutions emanating from any initial datum ◦ (α) (α) (α) u∈ Xm we observe that since Xm+jα is dense in Xm , finite blow-up of such a solution would contradict the theorem on continuous dependence of solutions on the initial data. ⊓ ⊔ Example 8. Suppose we have a(x) = xj , j > 0, and k(x, y) = xβ + y β . Then we can write k(x, y) = a(x)β/j + a(y)β/j so that α = β/j. The assumption for local solvability require α < 1; that is β < j while for the global solvability we additionally need αj ≤ 1; that is β ≤ 1. On the other hand, if k(x, y) = xβ y β , then the conditions of the local solvability remain the same while from 2xβ y β ≤ x2β + y 2β , it follows that we require β ≤ 1/2. References 1. Ch.D. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press, Inc., Orlando, 1985. 2. T. Apostol, Mathematical Analysis, Addison-Wesley, Reading, 1957. 3. W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc. (3) 54 (1987), 321–349. 4. W. 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