EVENTUAL COMPACTNESS FOR SEMIFLOWS GENERATED BY NONLINEAR AGE-STRUCTURED MODELS. P. Magal. H.R. Thieme. (Communicated by Michel Langlais)

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1 COMMUNICATIONS ON Website: PURE AND APPLIED ANALYSIS Volume 3, Number 4, December 24 pp EVENTUAL COMPACTNESS FOR SEMIFLOWS GENERATED BY NONLINEAR AGE-STRUCTURED MODELS P. Magal Department of Mathematics, Université du Havre, 7658 Le Havre, France H.R. Thieme Department of Mathematics, Arizona State University Tempe, AZ , USA (Communicated by Michel Langlais) Abstract. In this paper we investigate compactness properties for a semiflow generated by a semi-linear equation with non-dense domain. We start with the non-homogeneous linear case, and, and we derive some abstract conditions for non-autonomous semilinear equations. Then we investigate a special situation which is well adapted for age-structured equations. We conclude the paper by applying the abstract results to an age-structured models with an additional structure. 1. Introduction. In this paper, we consider an age-structured population model of the form ( t + a )u(t, a) = A(a)u(t, a) + J( t, u(t, ) ) { ( (a) +B 2 t, a, u2 (t) ) t >, u(t, a), < a < c, u(t, ) = u j (t) = c c B 1 (t, a, u 1 (t))u(t, a)da, t >, C j (t, a)u(t, a)da, t >, u(, a) = u (a), a < c, u(t, a) =, t, a > c. (1.1) The number c denotes the maximum possible age and u(t, ) is the age distribution of the population. The population may carry an additional structure which is coded in a Banach space Y with Y u(t, a). We refer to the books by Webb [45], Metz and Dieckmann [3], and Iannelli [23], Busenberg and Cooke [15], and Anita [7] for nice surveys on age-structured models. To investigate such a system, one can use solutions integrated along the characteristics, and derive a nonlinear Volterra equation. One can also use nonlinear semigroup theory. We refer to Webb [45] for more information about these two approaches. Here we use integrated semigroup theory to study equation (1.1). This paper is in the line of the works by Thieme [37, 4, 41, 39], Matsumoto, Oharu, 1991 Mathematics Subject Classification. 34Gxx, 35B41, 47H2, 47J35, 58D25, 92D25. Key words and phrases. Age structured models, attractors, asymptotically smooth semiflow. work partially supported by NSF grants DMS and

2 696 P. MAGAL AND H.R. THIEME Thieme [29], Magal [26], Thieme and Vrabie [42]. One can note that such a technique was also developed in the context of (neutral) delay differential equations. We refer to Adimy [1, 2], Adimy and Arino [3], Adimy and Ezzinbi [4, 5] for more precisions about the delay case. As we will see in Section 6, under some assumptions on the family of (eventually unbounded) linear operator A(a), one can find a Banach space X, a linear operator A : D(A) X X, and a map F : [, T ] D(A) X such that (1.1) can be written as U (t, s) x = x + A s U (t, s) xds + s F (s, U (t, s) x)ds, t s, x D(A). (1.2) The goal of the paper is to investigate compactness properties of the semiflow generated by (1.2). Under some conditions on F one has the uniqueness of the solutions of (1.2), and the family of nonlinear operator {U(t, s)} t s define a non-autonomous semiflow, that is to say that U(t, t) = Id, t, and U(t, r)u(r, s) = U(t, s), t r s. If we assume that for some bounded set B D(A), and some s, U(t, s)x exists for all t s, and for all x B. Then we look for conditions to verify that α (U(t, s)b), as t +, where α (.) is the measure of non-compactness of Kuratovski ( or ball measure of non-compactness) (see Martin [28], and Deimling [19] for more information about measures of non-compactness). In other words, we want to prove that the semiflow is asymptotically smooth in the sense of Hale [2]. By using this property one is then able to apply attractor theory. We refer to Hale [2], Temam [36], Babin and Vishik [11], Robinson [34], Ladyzhenskaya [25], Sell and You [35], Zhao [46] for nice surveys on global attractor theory. The compactness properties of semiflows were first investigated in the dense domain case (i.e. D(A) = X), by Ball [12, 13], Dafermos and Slemrod [17], Pazy [32, 33], Webb [44], Henri [22], Haraux [21], Vrabie [43]. Here, the point is to prove similar results for the non-dense domain case (i.e. D(A) X). Let {T (t)} t denote the C -semigroup of linear operator generated by the part of A in D(A). In the case where {T (t)} t is compact (i.e. T (t) is compact for each t > ) the problem is well understood (see Bouzahir and Ezzinbi [14], and Theorem 3.9 of this paper). In the case of age-structured models the semigroup {T (t)} t is not compact, and the problem becomes more complicated to study. In fact in the noncompact case, we need some regularity condition for the maps t F (t, U(t, s)x) (see Assumption 3.3 b) of Theorem 3.7, and Assumption 4.1 c) of Theorem 4.3). Because of this, the problem becomes completely different compared with the compact case. Also, from the case Y = R in age-structured model, it is clear that additional conditions like Assumption 3.3 b) are necessary in the non-compact case. In Thieme [4] this problem is investigated for linear age structured systems, and for a general Banach space Y. In Magal [26] the case Y = R n, J =, and c (, + ) is investigated. One can also compare Assumption 4.1 b) of this paper, and Assumption 5.3 d) in Magal [26], to see that the condition given here is more general. In particular with the condition given in Magal [26], one needs

3 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 697 much stronger assumptions in the applications to obtain some compactness properties (because one cannot use the technics developed in section 5 of this paper). In Thieme and Vrabie [42] this problem is investigated for a general Banach space Y, but they assume that J = and B 2 =. Here we investigate the general situation. In the general case, one needs to work more to derive some time regularity of t F (t, U(t, s)x) (compare section 3 in [42], and section 5 of this paper). For age structured problems, we also refer to Webb [45] for a nice treatment of this problem in the case where Y = R n, c = +, and by using solutions integrated along the characteristics. The plan of the paper is the following. In section 2, we recall some classical results about integrated semigroups. In section 3, we present some general compactness results for an abstract non-homogeneous Cauchy problem. The main novelty in section 3 is Theorem 3.7. Just for comparison between the case where {T (t)} t is compact, and the non-compact case, we also prove Theorem 3.9 which corresponds to the compact case. This will help the reader to compare both situations. In section 4, we derive a general result for non-autonomous semilinear systems. In section 5, we mainly investigate the regularity of t F (t, U(t, s)x), and we derive some abstract conditions that will be more applicable in the context of age structured systems. Finally in section 6 we apply the main result of section 5 to equation (1.1). 2. Preliminaries. In this section, we recall some classical results about integrated semigroups. We refer to Arendt [8][9], Kellermann and Hieber [24], Neubrander [31], Arendt et al. [1], and Thieme [38] for nice surveys on the subject. Let Y, Z be two Banach spaces, in the sequel we denote by L(Y, Z) the space of bounded linear operators from Y to Z. Assumption 2.1. Let A : D(A) X X be a linear operator. We assume that there exist real constants M 1, and ω R such that (ω, + ) ρ(a), and (λ A) n M, for all n N \ {}, and all λ > ω. (λ ω) n In the sequel, a linear operator A : D(A) X X satisfying Assumption 2.1 will be called a Hille-Yosida operator. We set X = D(A), and we denote by A the part of A in X, that is A x = Ax for all x D(A ) = {y D(A) : Ay X }. Then D(A ) is dense in X, and A generates a strongly continuous semigroup of linear operators on X that is denoted by {T (t)} t. Definition 2.2. A family of bounded linear operators S(t), t, on a Banach space X is called an integrated semigroup if and only if i) S() =. ii) S(t) is strongly continuous in t. iii) S(r)S(t) = r (S(l + t) S(l))dl = S(t)S(r) for all t, r. The generator A of a non-degenerate integrated semigroup is given by requiring that, for x, y X, x D(A), y = Ax S(t)x tx = S(s)yds t.

4 698 P. MAGAL AND H.R. THIEME It follows from this definition that S(t)x = A S(s)xds + tx t, x X. Notice that the previous formula implies that S(s)xds D(A), t, x X. So in particular S(t)x D(A), t, x X. It is well known that a Hille-Yosida operator A generates an integrated semigroup {S(t)} t L(X, X ). The family {S(t)} t is locally Lipschitz continuous. More precisely, we have for all t and s such that t s, S(t) S(s) M L(X,X) e ωr dr. The map t S(t)x is continuously differentiable if and only if x D(A), d dt S(t)x = T (t)x, t, x X, and T (r)s(t) = S(t + r) S(r) r, t. We also have the following explicit formula (see Magal [27]) for all x X, and for all µ > ω, S(t)x = µ T (s) (µ A) 1 xds + (µ A) 1 x T (t) (µ A) 1 x. (2.1) The main tool for nonlinear considerations is the following theorem which was first proved by Da Prato and Sinestrari [18] by using a direct approach, and by Kellermann and Hieber [24] by using integrated semigroups. Theorem 2.3. Assume that A is a Hille-Yosida operator, and f L 1 ((, τ), X). We set (S f)(t) = S(t s)f(s)ds, t [, τ]. Then t (S f)(t) is continuously differentiable, (S f)(t) D(A), t [, τ], t A(S f)(t) is continuous, and if we set u(t) = d dt (S f)(t), then and u(t) = A From now on, we define u(t) M u(s)ds + s f(s)ds, t [, τ], e ω(t s) f(s) ds. (S f) (t) := d (S f)(t). dt One can prove (see Thieme [37]) the following approximation formula (S f) (t) := lim λ + T (t s)λ (λi A) 1 f(s)ds, t [, τ]. From this approximation formula, we deduce that for all t and δ such that δ t τ, (S f) (t) T (δ) (S f) (t δ) = (S f(t δ +.)) (δ). (2.2)

5 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 699 Consider now the non-homogeneous Cauchy problem { du(t) = Au(t) + f(t), t ; dt u() = x X. (2.3) Definition 2.4. : A continuous function u C ([, τ], X) is called an integrated solution of (2.3) if and only if u(t) = x + A u(s)ds + f(s)ds, for all t [, τ]. One can note that in the previous formula we implicitly assume that u(s)ds D(A), t [, τ]. So we must have u(t) X, t [, τ]. From Theorem 2.3, we deduce that u(t) = T (t)x + (S f) (t) is the unique integrated solution of (2.3), and [ ] u(t) M e ωt x + e ω(t s) f(s) ds, t [, τ]. 3. Compactness for Non-homogeneous Problems. In this section we present some compactness results for a non-homogeneous Cauchy problem. More precisely we consider the following Cauchy problem du = Au(t) + f(t), for t [, τ], and u() =, dt and we investigate the compactness properties of the following set {(S f)(t) : t [, τ], f F}, where F is a subset of C ([, τ], X). From now on, we assume that A : D(A) X X is a Hille-Yosida operator. We start by the classical situation where f(t) belongs to X = D(A). Assumption 3.1. Let F C ([, τ], X ) be such that the subset {f(t) : t [, τ], f F} is bounded, and there exists δ (, τ), such that for each δ (, δ ), the subset is relatively compact. {T (δ)f(s) : s [δ, τ], f F} The following Theorem summarizes ideas from Webb [44]. Theorem 3.2. Let Assumption 3.1 be satisfied. Then is relatively compact. Proof. We set v f (t) := (S f)(t) = {(S f)(t) : t [, τ], f F} T (t s)f(s)ds, t [, τ], f F.

6 7 P. MAGAL AND H.R. THIEME Let δ (, δ ), be fixed. We have for each t [δ, τ], and each f F, T (t s)f(s)ds = t δ δ + T (t s)f(s)ds + T (t s)f(s)ds. δ By Assumption 3.1, for each δ (, δ ), the set M 1δ := {T (δ)f(t) : t [δ, τ], x E} δ T (t δ s)t (δ)f(s)ds (3.1) is compact. Moreover, as the map (t, x) T (t)x is continuous from [, + ) X into X, we deduce that for each δ (, τ), the set M 2δ := {T (t)x : t [, τ], x M 1δ } is compact. Therefore, for each δ (, τ) and each t [δ, τ], δ δ T (t δ s)t (δ)f(s)ds [, τ] co (M 2δ ) =: M 3δ, where co(m 2δ ) is the closed convex hull of M 2δ. By Mazur s theorem, M 3δ is compact. For each δ (, τ), we set and M δ = M 3δ {}, M = {v f (t) : x E, t [, τ]}. Then by using equation (3.1), the fact that M δ, and the fact that {f(t) : t [, τ], f F} is bounded, we deduce that there exists k >, such that for each δ (, τ), x M, y M δ, such that x y kδ. (3.2) Let ε > be fixed, and let δ > be fixed such that kδ ε/2. Since M δ is compact, we can find a finite sequence {y j } j=1,...,p such that ( M δ j=1,...,p B y j, ε ), 2 and by (3.2) we also have M j=1,...,p B (y j, ε). So M is relatively compact. The following lemma will be useful in section 5. Lemma 3.3. Let Assumption 3.1 be satisfied. Then t (S f)(t) is uniformly right continuous on [, τ), uniformly in f F. Proof. We set v f (t) := (S f)(t) = Let be t [, τ), h [, τ t), and f F. We have v f (t + h) v f (t) = = +h +h t T (t s)f(s)ds, t [, τ], f F. T (t + h s)f(s)ds T (t s)f(s)ds T (t + h s)f(s)ds + (T (h) Id) T (t s)f(s)ds.

7 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 71 So v f (t + h) v f (t) kh + (T (h) Id) T (t s)f(s)ds where k = Me ω+τ sup f F,t [,τ] f(t), and ω + = max(, ω). By Theorem 3.2, the subset { } C = T (t s)f(s)ds : t [, τ], f F is relatively compact. The map (t, x) T (t)x is continuous from [, + ) X into X, so this map is uniformly continuous on [, τ] C, and the result follows. We now consider the case where f(t) belongs to X. Assumption 3.4. a) Let be F C ([, τ], X). We assume that there exists λ > ω, such that the subset { (λ A) 1 f(t) : t [, τ], f F } is relatively compact. b) For each τ 1 (, τ), we assume that τ1 lim sup h f F f(s) 1 h s+h s f(l)dl ds =. Lemma 3.5. Let Assumption 3.2 be satisfied. Then, for each τ 1 (, τ), the set is relatively compact. Proof. We set v f (t) := (S f)(t), {(S f)(t) : t [, τ 1 ], f F} t [, τ], f L 1 ((, τ), X). Let τ 1 (, τ) be fixed. For each h (, τ τ 1 ), we define K h : L 1 ((, τ), X) C([, τ 1 ], X) by K h (f)(t) = 1 h +h t f(s)ds, t [, τ h]. Then d dt K h(f)(t) = 1 [f(t + h) f(t)], t [, τ h], f C([, τ], X). h For all t [, τ 1 ], and f C([, τ], X), we have so v Kh (f)(t) = (S K h (f))(t) = d dt v Kh (f)(t) = S(t)K h (f)() + S(s)K h (f)(t s)ds, S(s) 1 [f(t s + h) f(t s)] ds. (3.3) h By using equations (2.1), (3.3), Assumption 3.2 a), and by Mazur s theorem, we deduce that A τ1,h = { v Kh (f)(t) : t [, τ 1 ], f F } is a compact subset. We set A τ1, = {v f (t) : f F, t [, τ 1 ]}.

8 72 P. MAGAL AND H.R. THIEME By Theorem 2.3, for each h (, τ τ 1 ), and each t [, τ 1 ], we have v f (t) v Kh (f)(t) = (S f K h (f)) (t) t M e ω(t s) f(s) 1 s+h f(l)dl h s ds τ1 Me ω+ τ f(s) 1 s+h f(l)dl h ds, where ω + = max(, ω). So by using Assumption 3.2 b), we deduce that, for each ε >, there exists h ε (, τ τ 1 ) such that x A τ1,, y A τ1,h ε : x y ε. By using the same arguments as in the proof of Theorem 3.2, we conclude that A τ1, is relatively compact. Assumption 3.6. a) Let F C ([, τ], X) be such that {f(t) : t [, τ], f F} is a bounded set, and assume that there exists λ > ω, such that for each δ (, τ), the subset { (λ A) 1 f(t) : t [δ, τ], f F } is relatively compact. b) For each τ 1 (, τ), and each δ (, τ 1 ), we assume that τ1 lim sup h f F f(s) 1 s+h f(l)dl ds =. h δ The main result of this section is the following theorem. Theorem 3.7. Let Assumption 3.3 be satisfied. Then, for each τ 1 (, τ), the set is relatively compact. {(S f)(t) : t [, τ 1 ], f F} Remark: In applications, the main difficulty is to verify Assumption 3.3 b). It is clear that if F is a family of equicontinuous maps, then Assumption 3.3 b) is satisfied. Moreover, if F is a bounded set in W 1,1 ((, τ), X), then one can prove that Assumption 3.3 b) is satisfied. In section 5, we will study this question for a class of semi-linear problem. Proof. Let τ 1 (, τ) be fixed. We set v f (t) = (S f) (t), t [, τ 1 ], f F, and α 1 = sup f F,t [,τ] f(t). For each δ (, τ 1 ), and each f F, let v δf : [δ, τ] X be the unique solution of v δf (t) = A δ v δf (s)ds + By using Lemma 3.5, we deduce that s δ f(s)ds, t [δ, τ]. C δ = {v δf (t) : t [δ, τ 1 ], f F} {} is compact. s

9 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 73 By using Theorem 2.3, we have v f (t) Me ω+τ α 1 δ, t [, δ]. where α 2 = Me ω+τ α 1, and ω + = max(ω, ). Moreover, for t δ, we have v f (t) v δf (t) = v f (δ) + A so by using again Theorem 2.3, we obtain for t δ, We set δ v f (s) v δf (s)ds v f (t) v δf (t) Me ω+τ v f (δ) Me ω+τ α 2 δ. C = {v f (t) : t [, τ 1 ], f F}. Since C δ, we deduce that for δ (, τ), x C, y C δ, such that x y α 3 δ, where α 3 = Me ω+τ α 2. The relative compactness of C follows. Assumption 3.8. Let F C ([, τ], X) be such that {f(t) : t [, τ], f F} is a bounded set, and assume that {T (t)} t is compact. The proof of the following Theorem is adapted from Bouzahir and Ezzinbi [14]. Theorem 3.9. Let Assumption 3.4 be satisfied. Then the set is relatively compact. {(S f)(t) : t [, τ], f F} Proof. From equation (2.2) we have for all t and δ such that < δ t τ, We set (S f)(t) = T (δ) (S f) (t δ) + (S f(t δ +.)) (δ). C = {(S f)(t) : t [, τ], f F}, and C δ = {T (δ) (S f) (t δ) : t [δ, τ], f F} {}. Then by using Theorem 2.3, we deduce that there exists k >, such that for all δ (, τ), x C, y C δ : x y kδ, and as C δ is compact for all δ (, τ), the result follows. 4. Compactness for the Semilinear Problem. From now on, we assume that X is a Banach space, and A : D(A) X X is a Hille-Yosida operator. We consider u x (t) = x + A u x (s)ds + s F (s, u x (s))ds, for t [, τ]. (4.1) where F : [, τ] D(A) X is a continuous map. Let Y and Z be two Banach spaces, and let Ψ : Y Z be a map. We will say that Ψ is compact, if Ψ maps bounded subsets of Y into relatively compact sets of Z. The following lemma is adapted from Thieme [4], Theorem 7, p:698.

10 74 P. MAGAL AND H.R. THIEME Lemma 4.1. Let Y be a Banach space, and Ψ : [, τ] D(A) Y be a continuous map satisfying: a) For each < s t τ, the map x Ψ(t, T (s)x) is compact, the map t Ψ(t, x) is continuous on [, τ], uniformly with respect to x on bounded sets of D(A), and for each C >, there exists K(C) >, such that Ψ(t, x) Ψ(t, y) K(C) x y, whenever t [, τ], x, y C. b) There exists a bounded subset E D(A), such that for each x E, (4.1) has a solution u x (t) on [, τ], and {F (t, u x (t)) : t [, τ], x E} is bounded. Then for each η (, τ], the subset {Ψ(t, u x (t)) : x E, t [η, τ]} has a compact closure. Proof. By Theorem 2.3, the set {u x (t) : x E, t [, τ]} is bounded. We set α = sup u x (t), and α 1 = sup F (t, u x (t)). t [,τ],x E t [,τ],x E Let η (, τ] be fixed. By equation (2.2) we have for all t and δ such that δ t τ, (S F (., u(.))) (t) T (δ) (S F (., u(.))) (t δ) = (S F (t δ +., u(t δ +.))) (δ). Then by Theorem 2.3, for each δ [, τ], and each t [δ, τ], we have (S F (t δ +., u(t δ +.))) (δ) M δ e ω(δ s) F (t δ + s, u(t δ + s)) ds δ Mα 1 e ωs ds. We set γ = Me ω+τ α 1, with ω + = max(, ω). Then we obtain (S F (., u(.))) (t) T (δ) (S F (., u(.))) (t δ) δγ, δ [, τ], t [δ, τ]. So, for each δ t τ, we have Ψ(t, T (t) x + T (δ) (S F (., u(.))) (t δ)) = Ψ(t, T (δ) [T (t δ) x + (S F (., u(.))) (t δ)]). By using Theorem 2.3, we deduce that for t δ, So T (t δ) x + (S F (., u(.))) (t δ) Me ω+ (t δ) [α + (t δ) α 1 ] Me ω+τ [α + τα 1 ] =: α 2. T (t) x + T (δ) (S F (., u(.))) (t δ) Me ω+τ α 2 =: α 3. (4.2) Moreover, since the map t Ψ(t, x) is continuous on [, τ], uniformly with respect to x on bounded sets of D(A), we deduce that for each δ (, τ), the set { } Ĉ δ = Ψ(t, T (δ)x) : x D(A), x α 3, t [δ, τ] is relatively compact. So by (4.2), for each δ (, τ), the set C δ = {Ψ(t, T (t) x + T (δ) (S F (., u(.))) (t δ)) : x E, t [δ, τ]} is compact. For each δ (, τ), let v x,δ (t) be the unique solution of v x,δ (t) = T (t) x + T (δ) (S F (., u(.))) (t δ), for t [δ, τ].

11 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 75 Then for each δ (, η), and each t [η, τ], we have Ψ(t, u x (t)) Ψ(t, v x,δ (t)) K (α 3 ) (S F (t δ +., u(t δ +.))) (δ) We set and We deduce that for each δ (, η), So C,η is relatively compact. K (α 3 ) γδ. C,η = {Ψ (t, u x (t)) : x E, t [η, τ]}, γ = K (α 3 ) γ. x C,η, y C δ, such that x y γδ. Assumption 4.2. We assume that F : [, τ] D(A) X is a continuous map, which satisfies F (t, x) = F 1 (t, x) + H(t, x)x + Γ(t, x), ( ) where F 1 : [, τ] D(A) X, H : [, τ] D(A) L D(A) and Γ : [, τ] D(A) D(A) are continuous maps, with the following: a) There exists a bounded set E X such that, for each x E, there exists a continuous solution u x : [, τ] X of (4.1) such that { F1 (t, u x (t)) : t [, τ], x E } {, Γ(t, ux (t)) : t [, τ], x E }, and { H(t, u x (t)) : t [, τ], x E } are bounded sets. b) There exists λ > ω, such that for each δ (, τ), the set { } (λ A) 1 F 1 (t, u x (t)) : t [δ, τ], x E, is relatively compact. c) For each τ 1 (, τ), and each δ (, τ 1 ), τ1 lim sup h x E F 1(s, u x (s)) 1 h δ s+h s F 1 (l, u x (l))dl ds =. d) The map t Γ(t, x) is continuous from [, τ] into D(A), uniformly with respect to x in bounded subsets of D(A), and for each t, s [, τ], with s >, the map x T (s) Γ(t, x) is compact from D(A) into D(A). e) For each δ (, τ), the set {H(t, u x (t)) : t [δ, τ], x E} is relatively compact. f) We assume that there exists < τ τ < τ, such that for each x E, if u 3x C ([, τ], X ) is the solution of u 3x (t) = T (t)x + T (t s)h(s, u x (s))(u 3x (s))ds, t [, τ], then the subset {u 3x (t) : t [τ, τ ], x E} is relatively compact.

12 76 P. MAGAL AND H.R. THIEME From now on, for each x E, and each t [, τ], we define u 1x (t) = (S F 1 (., u x (.))) (t) + u 2x (t) = u 3x (t) = T (t)x + T (t s)γ(s, u x (s))ds + By uniqueness of the solution of v(t) = x + A we deduce that v(s)ds + T (t s)h(s, u x (s))(u 1x (s))ds T (t s)h(s, u x (s))(u 3x (s))ds. T (t s)h(s, u x (s))(u 2x (s))ds F 1 (s, u x (s)) + Γ(s, u x (s)) + H(s, u x (s))(v(s))ds, u x (t) = u 1x (t) + u 2x (t) + u 3x (t), t [, τ], x E. The main result of this section is the following theorem. Theorem 4.3. Let Assumptions 4.1 a)-e) be satisfied. Then for each τ 1 (, τ), the set {u 1x (t) + u 2x (t) : t [, τ 1 ], x E} has compact closure. If in addition Assumption 4.1 f) is satisfied then the set has compact closure. {u x (t) : t [τ, τ ], x E} Remark: 1) One can use Lemma 4.1 to verify Assumptions 4.1 b) and e). One can also relax the Lipschitz condition in Lemma 4.1 by using similar idea as in Lemma ) In the applications, the main difficulty is to verify Assumption 4.1 c). The section 5 is devoted to this question. 3) The component u 3x (t) corresponds to the non-compact part of the semiflow. Also the first part of Theorem 4.3 can be used to prove the existence of a global attractor when the semiflow is not eventually compact, but contracting for some measure of non-compactness (see Sell and You [35] for a definition of contracting semiflows). Proof. First by using Assumption 4.1 a), Theorem 2.3, and Gronwall s lemma, we deduce that {u x (t) : t [, τ], x E} is a bounded set. Moreover by taking into account Assumption 4.1 f), it remains to prove the compactness of {u ix (t) : t [, τ 1 ], x E}, for i = 1, 2, τ 1 (, τ). Let τ 1 (, τ) be fixed. By Assumptions 4.1 a)-c), and Theorem 3.7, the set {(S F 1 (., u x (.))) (t) : t [, τ 1 ], x E} is relatively compact. By Assumption 4.1 d), for each δ (, τ), the set {T (δ)γ(t, u x (t)) : t [, τ], x E} is relatively compact. So by using Assumption 4.1 a), and Theorem 3.2, we deduce that { } T (t s)γ(s, u x (s))ds : x E, t [, τ]

13 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 77 has a compact closure. We set for each x E, and each t [, τ], v 1x (t) = (S F 1 (., u x (.))) (t), and v 2x (t) = For i = 1, 2, we obtain that where L x (ψ(.))(t) = u ix (t) = L k x(v ix (.))(t), k= So for each integer m 1, m u ix (t) = L k x((v ix ) (.))(t) + k= T (t s)h(s, u x (s))(ψ(s))ds. k=m+1 T (t s)γ(s, u x (s))ds. L k x((v ix ) (.))(t), t [, τ]. By using Assumption 4.1 e), we deduce that for each δ (, τ), the set M δ := {H(s, u x (s))(v ix (s)) : x E, s [δ, τ 1 ]} is relatively compact. So by using Theorem 3.2 we deduce that for each i = 1, 2, there exists a compact set C i D(A) such that L x (v ix (.))(t) C i, t [, τ 1 ]. By using induction arguments we deduce that for each m 1, and each i = 1, 2, there exists a compact subset Cm i X, such that m L k x(v ix (.))(t) Cm, i t [, τ 1 ]. Moreover, we have k=1 L k x((v ix ) (.))(t) γ i e ω+τ α k τ k k!, where M 1, ω IR the constants from Theorem 2.3, ω + : = max(, ω), γ i := sup v ix (t), t [,τ],x E and α : = M sup H(t, u x (t)) L(D(A)). t [δ,τ],x E We deduce that t [, τ], L k x (v ix (.))(t) γ i e ω+ τ k=m+1 and γ m as m +. We let k=m+1 (ατ) k k! γ i e ω+τ (e ατ C i = t [,τ1],x E {u ix (t)} for i = 1, 2. m (ατ) k ) =: γ m, k! Then for all x C i there exists y C i m such that x y γ m. So for i = 1, 2, C i is relatively compact. k=

14 78 P. MAGAL AND H.R. THIEME 5. More about the Semi-linear Case. In this section we derive abstract conditions which imply in particular Assumptions 4.1 c). Consider the equation u x (t) = x + A u x (s)ds + F (s, u x (s))ds, for t [, τ]. (5.1) Assumption 5.1. There exists a bounded set E X such that, for each x E, there exists a continuous solution u x : [, τ] X of (5.1), F : [, τ] D(A) X is a continuous map, which satisfies F (t, x) = F 1 (t, x) + H(t, x)x + Γ(t, x), ( ) where F 1 : [, τ] D(A) X, H : [, τ] D(A) L D(A) and Γ : [, τ] D(A) D(A) are continuous maps, satisfying the following: a) The maps t F 1 (t, x), t Γ(t, x), t H(t, x) are continuous on [, τ], uniformly with respect to x in bounded sets of D(A). b) There exists a bounded set E X such that, for each x E, there exists a continuous solution u x : [, τ] X of (5.1), and the sets {F 1 (t, u x (t)) : t [, τ], x E}, {Γ(t, u x (t)) : t [, τ], x E}, and {H(t, u x (t)) : t [, τ], x E}, are bounded. As in section 4, for each x E, and each t [, τ], we define u 1x (t) = (S F 1 (., u x (.))) (t) + u 2x (t) = u 3x (t) = T (t)x + T (t s)γ(s, u x (s))ds + T (t s)h(s, u x (s))(u 1x (s))ds T (t s)h(s, u x (s))(u 3x (s))ds. T (t s)h(s, u x (s))(u 2x (s))ds Lemma 5.2. Let Assumption 5.1 be satisfied. Then {u x (t) : t [, τ], x E}, and {F (t, u x (t)) : x E, t [, τ]} are bounded sets. Proof. The result follows from Assumption 5.1 b), and Gronwall s lemma. Assumption 5.3. For all c, ε >, and all t [, τ), there exist n N, bounded linear operator H j from D(A) into Banach spaces Z j, 1 j n, and continuous maps G j from Z j into a Banach spaces Y j, such that the following holds: H(t, x) H(t, x) n j=1 G j (H j x) G j (H j x) + ε whenever x, x D(A), x, x c. For each j = 1,..., n, H j T (t) is compact for t >. The following lemma allows to suppress the Lipschitz condition of Lemma 4.1. Lemma 5.4. Let Assumptions 5.1 and 5.2 be satisfied. Then, for each δ (, τ], the set {H(t, u x (t)) : t [δ, τ], x E}, is a relatively compact set.

15 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 79 Proof. Let δ (, τ] be fixed. Let {t k } k [δ, τ] and {x k } k E be two sequences. We define y k = H(t k, u xk (t k )), k. It is sufficient to show that for each ε >, we can extract a sequence { y kp }p, such that there exists p, such that ykp y kl ε, p, l p. Then by setting ε = 1 j+1, j IN, and by diagonalization procedure, we can extract a converging subsequence. Let ε > be fixed. Since {t k } k [δ, τ], we can extract a subsequence (for which we use the same index) such that t k t [δ, τ], as k +. Since {F (t, u x (t)) : x E, t [, τ]} is bounded, there exists α >, such that We have k, l, u x (t) α, t [, τ], x E. H(t k, u xk (t k )) H(t l, u xl (t l )) H(t k, u xk (t k )) H( t, u xk (t k )) + H( t, u xk (t k )) H( t, u xl (t l )) + H( t, u xl (t l )) H(t l, u xl (t l )), and since t H(t, x) is continuous on [, τ], uniformly with respect to x in bounded sets of D(A), we can find k, such that for all k, l k, H(t k, u xk (t k )) H(t l, u xk (t l )) ε 2 + H( t, u xk (t k )) H( t, u xl (t l )). Choose n IN, operators H j, and maps G j according to Assumption 5.2, for ε 4 rather than ε, t = t (, τ), and c = α. Then H( t, u xk (t k )) H( t, u xl (t l )) n G j (H j u xk (t k )) G j (H j u xl (t l )) + ε 4. j=1 By Lemma 4.1 (with Y = Z j, and Ψ(t, x) = H j x), we deduce that for each j = 1,..., n, {H j u x (t) : t [δ, τ], x E} is relatively compact. So, we can find a converging subsequence of {H j u xk (t k )} k (that is denoted with the same index), and H j u xk (t k ) z j as k +, for each j = 1,..., n. For each j = 1,..., n, we have G j (H j u xk (t k )) G j (H j u xl (t l )) G j (H j u xk (t k )) G j (z j ) + G j (H j u xl (t l )) G j (z j ) and since G j is continuous, we can find k 1, such that for each k, l k 1, and each j = 1,..., n, G j (H j u xk (t k )) G j (H j u xl (t l )) ε 4n. Finally, for all k, l k 1, we obtain H(t k, u xk (t k )) H(t l, u xl (t l )) ε. Assumption 5.5. For all t, s [, τ], s >, the map x T (s) Γ(t, x) is compact from D(A) into D(A). Lemma 5.6. Let the Assumptions be satisfied. Then {u 2x (t) : t [, τ], x E} is a relatively compact set, and the map t u 2x (t) is right continuous, uniformly with respect to x E.

16 71 P. MAGAL AND H.R. THIEME Proof. By using Lemma 5.4, and the same arguments as in the proof of Theorem 4.3, we deduce that {u 2x (t) : t [, τ], x E} is a relatively compact set. By using again Lemma 5.4, we deduce that for each δ (, τ], the set {H(t, u x (t))u 2x (t) : t [δ, τ], x E} is a relatively compact set. Since we have u 2x (t) = Γ(t, u x (t)) + H(t, u x (t))(u 2x (t)) X, t [, τ], x E, T (t s) [Γ(s, u x (s)) + H(s, u x (s))(u 2x (s))] ds, t [, τ], and the result follows from Lemma 3.3. Assumption 5.7. F 1 (t, x) is of the form F 1 (t, x) = K 1 G(t, x), with X 1 a Banach space, G : [, τ] D(A) X 1 a continuous map, K 1 : X 1 X a bounded linear operator, with the following: c) There exists λ > ω, such that for each δ (, τ), the set { } (λ A) 1 F 1 (t, u x (t)) : t [δ, τ], x E, is relatively compact. ( ) d) There exists a Banach space X,. X b, with X D(A) such that, for all c, ε > and all t [, τ), there exist n N, bounded linear operators H j from D(A) into Banach ) spaces Y Hj, 1 j n, and continuous maps L j from Y Hj into L (D(A), Y Lj, the space of bounded linear operators from D(A) into a Banach spaces Y Lj, such that the following holds: 1. F 1 (t, x) F 1 (t, x) n j=1 L j (H j x) x L j (H j x) x + ε whenever x, x D(A), x, x c. 2. H j T (t) is compact from D(A) to Y Hj for all j = 1,..., n, t >. 3. If K {Hj : j = 1,..., n} { L j (z) : z Y Hj, j = 1,..., n }, then KT (t) L( X, Y K e ) for all t >, and t KT (t) is operator norm continuous from (, + ) into L( X, Y K e ), and we have for all sufficiently large λ > that K (λ A) 1 K 1 = e λs W (s)ds on X, with W (t) forming an exponentially bounded operator-norm Borel measurable family of bounded linear operators. e) There exists τ [, τ) such that x E, t [τ, τ], and (u 1x + u 3x ) (τ ) X, and H(t, u x (t)) (u 1x + u 3x ) (t) X, sup x E,t [τ,τ] sup (u 1x + u 3x ) (τ ) X b < +, x E H(t, u x (t)) (u 1x + u 3x ) (t) X b < +.

17 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 711 Remark: If for each K {H j : j = 1,..., n} { L j (z) : z Y Hj, j = 1,..., n }, t KT (t) is operator norm continuous from (, + ) into L(D(A), Y e K ), then we can choose X = D(A) and τ =. But in the example of section 6, we will need to choose a Banach X D(A), to obtain such operator norm continuity. We now prove that Assumption 4.1 c) is satisfied. Lemma 5.8. Let Assumptions 5.1 and 5.4 be satisfied. Further let K 2 be a bounded linear operator from X into a Banach X 2 such that, for sufficiently large λ >, K 2 (λ A) 1 K 1 = + e λt W (t)dt on X 1, with W (t) forming an exponentially bounded operator-norm Borel measurable family of bounded linear operators. Assume that K 2 T (t) L( X, X 2 ), t >, and t K 2 T (t) is operator norm continuous from (, + ) into L( X, X 2 ). Then for each δ (, τ τ ), the map t K 2 (u x (t)) is uniformly right-continuous on [τ + δ, τ), uniformly in x E. Proof. Let δ (, τ τ ) be fixed. We set v x (t) = u 1x (t) + u 3x (t), x E, t [, τ]. By taking into account Lemma 5.6, it is sufficient to show that t K 2 (v x (t)) is uniformly right-continuous on [τ + δ, τ), uniformly in x E. We have for each t [, τ τ ], and each x E, v x (t + τ ) = T (t)v x (τ ) + S F 1 ( τ +, u x (τ + ) ) (t) + T (t s)h(τ + s, u x (τ + s))(v x (τ + s))ds The uniqueness property of the Laplace transform implies that for all t, K 2 S(t)K 1 = So for each t [, τ τ ], and each x E, K 2 v x (t + τ ) = V (t)v x (τ ) + + V (t s)f x (s)ds, W (s)ds on X 1. W (t s)g(τ + s, u x (τ + s))ds where f x (t) = H(τ + t, u x (τ + t))(v x (τ + t)), V (t) = K 2 T (t) for all t. Since V (t) is operator-norm continuous from (, + ) into L( X, X 2 ), and { v x (τ ) b X : x E } is bounded, we deduce that t V (t)v x (τ ) is uniformly continuous in t on [δ, τ τ ], uniformly in x E. So it remains to consider t [, τ τ ], l x (t) = and k x (t) = V (t s)f x (s)ds, W (t s)g(τ + s, u x (τ + s))ds.

18 712 P. MAGAL AND H.R. THIEME For Y = X or Y = X, we set C Y = sup { f x (t) Y : x E, t [, τ τ ]}. Let ε > be fixed, let be t [δ, τ τ ), and h (, τ τ t). Then +h l x (t) l x (t + h) X2 V (t s)f x (s)ds V (t + h s)f x (s)ds X2 +h V (t + h s)f x (s)ds + [V (t s) V (t + h s)] f x (s)ds. X2 X2 t In addition, let γ (, δ). Then l x (t) l x (t + h) X2 C X h K 2 Me ωτ + [V (t s) V (t + h s)] f x (s)ds C X h K 2 Me ωτ + [V (t s) V (t + h s)] f x (s)ds t γ X 2 t γ + [V (t s) V (t + h s)] f x (s)ds X2 C X h K 2 Me ωτ + 2C X γ K 2 Me ωτ t γ + [V (t s) V (t + h s)] fx (s) X2 ds. C X h K 2 Me ωτ + 2C X γ K 2 Me ωτ + (t γ) sup s [,t γ] V (t s) V (t + h s) L( b X,X2) sup s [,t γ] f x (s) b X If we choose γ > small enough, we have l x (t) l x (t + h) ε 2 + C Xh K 2 Me ωτ + tc b X sup V (l) V (l + h) L( b. X,X2) γ l t But t V (t) is operator norm continuous from [γ, τ τ ] into L( X, X 2 ), so it is uniformly operator norm continuous on [γ, τ τ ]. We deduce that l x (t + h) l x (t) as h, uniformly in t [δ, τ τ ), x E. We now consider k x (t). For t [, τ), and < h τ t, k x (t + h) = = W (t + h s)g(s, u x (s))ds + W (h + s)g(t s, u x (t s))ds + So for some M >, k x (t + h) k x (t) M τ h +h t W (t + h s)g(s, u x (s))ds W (h s)g(t + s, u x (t + s))ds. W (h + s) W (s) ds + M Since W (t) is Borel measurable with respect to operator-norm, we have h W (s) ds. k x (t + h) k x (t) as h, uniformly in t [, τ), x E. X2 Proposition 5.9. Let Assumptions be satisfied. Then, for each t [, τ] and each δ (, τ τ ), s F 1 (t, u x (s)) is uniformly right-continuous on [τ + δ, τ], uniformly in x E.

19 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 713 Proof. Let α > such that u x (t) α, t [, τ], x E. Let ε >, t [, τ]. Choose n N and operators H j and maps L j according to Assumption 5.4 d) for ε/2 rather than ε, and c = α. Set G j (x) = L j (H j x)x. Obviously it is sufficient to show that, for any δ (, τ τ ), G j u x is uniformly right continuous on [τ + δ, τ], uniformly with respect to x E. Let K be an operator in {H j : j = 1,..., n} { L j (z) : z Y Hj, j = 1,..., n }. By Lemma 5.8, for each δ (, τ τ ), t Ku x (t) is uniformly right-continuous on [τ + δ, τ], uniformly with respect to x in E. Let δ (, τ τ ), and j {1,..., n} be fixed. By Assumption 5.4 d)-2), H j T (t) is a compact operator for t >. So by Lemma 4.1 (with Y = Y Hj, and Ψ(t, x) = H j x), the subset M δj = {H j u x (s) : s [τ + δ, τ], x E} is compact. Suppose that G j u x is not uniformly continuous on [τ + δ, τ), uniformly with respect to x E. Then there exist sequences (s k ) in [τ + δ, τ), (x j ) in E and (h k ) in (, 1) such that s k + h k < τ, h k as k, and lim inf k G j (u xk (s k + h k )) G j (u xk (s k )) >. After choosing a subsequence, s k s [τ + δ, τ]. Since M δ,j is compact, we can choose another subsequence such that H j u xk (s) y Y Hj. Since H j u xk is uniformly continuous on [τ + δ, τ] uniformly in k, H j u xk (s k ) y, H j u xk (s k + h k ) y, k. Since L j is continuous from Z j to L(D(A), Y j ), L j (H j u xk (s k )) L j (y), L(H j u xk (s k + h k )) L j (y), k, with the convergence holding in operator norm. Since t L j (y)u x (t) is uniformly right-continuous on [τ + δ, τ], uniformly with respect to x in E, we have By definition of G j, L j (y)u xk (s k + h k ) L j (y)u xk (s k ), k. G j (u xk (s k + h k )) G j (u xk (s k ) L j (H j u xk )u xk (s k + h k ) L j (y)u xk (s k + h k ) + L j (y)u xk (s k + h k ) L j (y)u xk (s k ) + L j (y)u xk (s k ) L j (H j u xk )u xk (s k ) L j (H j u xk ) L j (y) α + L j (y)u xk (s k + h k ) L j (y)u xk (s k ) + L j (y) L j (H j u xk ), k, and we obtain a contradiction. Corollary 5.1. Let Assumptions be satisfied. Then, if δ (, τ τ ), t F 1 (t, u x (t)) is uniformly right-continuous on [τ + δ, τ), uniformly in x E. Proof. Let δ (, τ τ ), and ε > be fixed. By Assumption 5.1, the map t F 1 (t, x) is uniformly continuous on [, τ], uniformly with respect to x in bounded sets. So, we can choose some η > such that F 1 (t, u x (s)) F 1 (r, u x (s)) < ε/4,

20 714 P. MAGAL AND H.R. THIEME whenever t r < η, r, s, t [, τ], x E. Further we choose a partition τ + δ = t t n+1 = τ such that t j t j 1 < η, j = 1,..., n + 1. Then, if < h < η and s [τ + δ, τ), with s + h τ, F 1 (s + h, u x (s + h)) F 1 (s, u x (s)) < ε/4 + F 1 (s, u x (s + h)) F 1 (s, u x (s)). For each s [τ + δ, τ) we find some j {1,..., n} such that s t j < η. Hence F 1 (s + h, u x (s + h)) F 1 (s, u x (s)) 3ε 4 + F 1(t j, u x (s + h)) F 1 (t j, u x (s)). By Proposition 5.9, for each j = 1,..., n, there exists some η j > such that F 1 (t j, u x (s + h)) F 1 (t j, u x (s)) ε/4 whenever < h < η j, s [τ + δ, τ), with s + h < τ, x E. Set η = min j=,...,n η j, then F 1 (t j, u x (s + h)) F 1 (t j, u x (s)) ε whenever < h < δ, s [τ + δ, τ), x E, s + h < τ. Assumption We assume that there exist τ, τ : τ < τ τ < τ, such that for each x E, if u 3x,τ C ([τ, τ], X ) is the solution of u 3x,τ (t + τ ) = T (t)u x (τ ) + T (t s)h(τ + s, u x (τ + s))(u 3x,τ (s + τ ))ds, t [, τ τ ], then the subset {u 3x,τ (t) : t [τ, τ ], x E} is relatively compact. Theorem Let the Assumptions be satisfied. Then for each τ 1 (τ, τ), the set {u 1x (t) + u 2x (t) : t [τ, τ 1 ], x E} has compact closure. If in addition Assumption 5.5 is satisfied then the set has compact closure. {u x (t) : t [τ, τ ], x E} Proof. We have for each x E, and each t [τ, τ], u x (t) = u x (τ ) + A τ u x (s)ds + τ F (s, u x (s))ds, so by replacing the initial time by τ, the result follows from Theorem Application to an Age-structured Population Model with an Additional Structure. Let Y be a Banach space that represents the distribution of a population with respect to a structure different from age, e.g., induced by space or body size. It can also represent the distributions of several populations with or without a structure different from age. The additional structure would then come from the multi-species composition. Let u(t, a) denote the structural distribution (with respect to this structure) of the individuals with age a at time t. More precisely u(t, ) L 1 (, c, Y ) where the latter denotes the space of integrable functions

21 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 715 on (, c) with values in Y. We consider the following model: ( t + a )u(t, a) = A(a)u(t, a) + J( t, u(t, ) ) { (a) ( +B 2 t, a, u2 (t) ) t >, u(t, a), < a < c, u(t, ) = c B 1(t, a, u 1 (t))u(t, a)da, t >, u j (t) = c C j(t, a)u(t, a)da, t >, u(, a) = u (a), a < c, u(t, a) =, t, a > c. (6.1) The number c (, ) denotes the maximum age of an individual. The operators A(a) describe how individuals of age a change with respect to the other structure and also to what extent they die a natural death (i.e. a death not depending on the distribution of the population(s)) or emigrate. J(t, x) can be interpreted as an immigration rate which depends on the number and age distribution of the resident population described by x L 1 (, c, Y ). The operators B 2 (t, a, u 1 (t)) may represent additional mortality factors that depend on the density (or densities) of the population(s). If the populations are counted as biomasses, also biomass gains through predation may be incorporated here. The boundary condition describes the birth of individuals. The operators B 1 (t, a, z) represent the per capita birth rates of individuals with age a where z is the competition by other individuals. The operators C 1 (t, a) may describe to what degree individuals compete or cooperate for the resources necessary for reproduction. Assumption 6.1. a) For j = 1, 2, let C j : [, τ] [, c) L(Y, Z j ) have the following properties: i) For every t [, τ], y Y, C j (t, a)y is Borel measurable in a [, c). ii) For every t [, τ], ess sup C(t, a) < and <a<c ess sup C(t, a) C(t, a), t t. <a<c iii) For every t [, τ], r [, c), and y Y, there exists a subset N = N t,r,y of [, c) with Lebesgue measure such that {C j (t, a)y; a [r, c) \ N} has compact closure in Z j. b) The map J : [, τ] L 1 (, c, Y ) L 1 (, c, Y ) is continuous. Further, if Ẽ is a bounded subset of L 1 (, c, Y ) and t >, J(t, v) is continuous in t [, τ], uniformly in v Ẽ, and the following hold: i) If b, b 1, b 2 [, c), b 2 > b 1, and b 2 b, b 1 b, then b 2 b 1 J(t, v)(a) da uniformly in v Ẽ. ii) J(t, v)(a) da as m, uniformly for v Ẽ. {a [,c); J(t,v)(a) >m} iii) For any η (, c s), η U(s + a + h, a + h)j(t, v)(a + h) U(s + a, a)j(t, v)(a) da, as h, uniformly in v Ẽ. c) For j = 1, 2, let B j : [, τ] [, c) Z j L(Y, Y ) have the following properties: i) For every t [, τ], y Y, z Z j, B j (t, a, z)y is Borel measurable in a [, c), and ess sup B j (t, a, z) <. <a<c

22 716 P. MAGAL AND H.R. THIEME ii) For every t [, τ], ess sup B j (t, a, z) B j (t, a, z ), z z <a<c and for every δ >, sup ess sup B j (t, a, z) B j (t, a, z), t t. z δ <a<c iii) For every t [, τ], z Z j, r [, c), and y Y, there exists a subset N = N t,z,r,y of [, c) with Lebesgue measure such that {B j (t, a, z)y; a [r, c) \ N} has compact closure in Z j. There are various assumptions and procedures under which the operators A(a) can be generators of an evolutionary system U(t, r) : Y Y, t r such that lim (U(t, t h)x x) = A(t)x, h h 1 t >, x D(A(t)). We will take the point of view that an evolutionary system U is given and interpret the operators A(a) in a generalized sense by looking at the associated evolution semigroup. Assumption 6.2. Let U(t, s), s t < c, be a family of bounded linear operators on Y with the following properties: a) U(t, s)u(s, r) = U(t, r), r s t < c. b) U(s, s)x = x, x Y, s < c. c) U(t, s)x is a continuous function of s and t for s t < c and x Y. d) sup s t<c U(t, s) <. e) U(t, s) is compact for each < s < t. f) For all r [, c), y Y, {U(a, r)y; a [r, c)} has compact closure in Y. Here X = Y L 1 ((, c), Y ), and X = {} L 1 ((, c), Y ). Let ( T (t)x =, T ) (t)v, x = (, v) X, { where T (t)} is the evolutionary semigroup on t L1 (, c, Y ) associated with U, defined for each v L 1 (, c, Y ), and almost every a (, c), by { ( T U(a, a t)v(a t), if a t (t)v)(a) =, if a < t. We refer to Chicone and Latushkin [16](and the references therein) for a nice overview about this type of semigroups. It is readily checked that T is a C - semigroup on L 1 (, c, Y ). By Assumption 6.2 d) there exists Λ 1, such that U(t, s) Λ, s t < c. It follows from the considerations in [41], Sec. 5, that there exists a Hille-Yosida operator A : D(A) X X such that (λ A) 1 (x, v) = (, w), w(a) = e λa U(a, )x + a e λs U(a, a s)v(a s)ds,

23 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 717 and there exists M 1, such that (λ A) n Mλ n, λ >, n IN. Then D(A) = X, and A generates an integrated semigroup S(.). We define H j : [, τ] X Z j, for j = 1, 2, for each x = (, v) X, and each t [, τ], by H j (t)x = c C j (t, a)v(a)da. We define F : [, τ] X X, F 1 : [, τ] X X, and H : [, τ] X L (X ), for each x = (, v), y = (, w) X, and each t [, τ], by ( c ) F 1 (t, x) = B 1 (t, a, H 1 (t) (, v))v(a)da,, H(t, x)y = (, B 2 (t, a, H 2 (t) (, v))w(a)), Γ(t, x) = (, J(t, v)), and F (t, x) = F 1 (t, x) + H(t, x)x + Γ(t, x). We now consider equation (6.1) under the following abstract form du x (t) = Au x (t) + F (t, u x (t)), t, u x () = x. (6.2) dt The main result of this section is the following theorem. Theorem 6.3. Let the Assumptions be satisfied and τ > 2c. Assume that there exists a bounded set E X such that for each x E, (6.2) has an integrated solution u x (t) on [, τ], the subset {F 1 (t, u x (t)) : t [, τ], x E} is bounded, sup t τ,x E ess sup B 2 (t, a, H 2 (t)(, u x (t))) <, <a<c c sup t τ,x E J(t, u x(t))(a) da <. Then for each τ [2c, τ), the set {u x (t) : x E, t [2c, τ]} has a compact closure. To prove the previous theorem we now apply Theorem In order to apply Theorem 5.12, we need some preliminary results. The following result will be used to verify Assumption 5.3. Lemma 6.4. Let M L 1 (, c, Y ), s >. Then T (s)m is compact if the following holds: i) If b, b 1, b 2 [, c), b 2 > b 1, and b 2 b, b 1 b, then b 2 b 1 f(a) da uniformly in f M. ii) f(a) da as m, uniformly for f M. {a [,c); f(a) >m} iii) For any η (, c s), η U(s+a+h, a+h)f(a+h) U(s+a, a)f(a) da as h, uniformly in f M. Proof. For f L 1 (, c; Y ), we define Ξ i (f)(a) = i a+ 1 i a U(s + r, r)f(r)dr, where U(s + r, r)f(r) := for r c s, further { f(a), f(a) m f [m] (a) =, f(a) > m a, i IN, }, m IN.

24 718 P. MAGAL AND H.R. THIEME Let i IN, 1 i < s 4. We want to apply the Arzela-Ascoli theorem to the sets {Ξ i (f) : f M}. By i) and by Assumption 6.2 d), {Ξ i (f) : f M} is an equicontinuous set. We claim that, for all a [, c), {Ξ i (f)(a) : f M} is a compact set in Y. To prove this claim, let a [, c). After a substitution, Ξ i (f [m] )(a) = 1 U(s + a + r i, a + r i )f [m](a + r i )dr. { Ξi (f [m] )(a) : f M } is contained in the closed convex hull of the set M = {U(s + a + r i, a + r i )f [m](a + r i ) : r [, 1], f M } {U(s + a + r i, s + a)u(s + a, s 2 + a)y : y N, r [, 1] } where N = {U( s 2 + a, a + r i )f [m](a + r i ) : r [, 1], f M }. Notice that s 2 > r i for all r [, 1] by our choice of i. By Assumption 6.2 d) and the definition of f [m], N is a bounded set. By Assumption 6.2 e), U(s + a, s 2 + a) is a compact operator and so N = U(s + a, s 2 + a)n has compact closure. Since the map (r, y) U(s + a + r i, s + a)y is continuous, { M U(s + a + r } i, s + a)y : y N, r [, 1] has compact closure. By a theorem by Mazur, the closed convex hull of M is compact and so is { Ξ i (f [m] )(a) : f M }. By definition of f [m], Ξ i (f)(a) Ξ i (f [m] )(a) = i U(s + r, r)f(r)dr. [a,a+ 1 i ] {r (,c): f(r) >m} By Assumption 6.2 d), there exists some constant Λ >, such that Ξi (f)(a) Ξ i (f [m] )(a) iλ f(r) dr {r (,c): f(r) >m}, m, uniformly for f M. Here we have used ii). Since { Ξ i (f [m] )(a) : f M } has compact closure, so has {Ξ i (f)(a) : f M}. In order to show that T (s)m has compact closure, let (f k ) be a sequence in M. It follows from our previous considerations and Arzela-Ascoli theorem that, for any i IN, there exists a subsequence of (Ξ i (f k )) k IN which converges locally uniformly on [, c s), i.e. uniformly on [, c s η] for every sufficiently small η >. By a diagonalization procedure, we have, after choosing a subsequence, that (Ξ i (f k )) k IN converges locally uniformly on [, c s) for every i IN. It follows from ii) and Assumption 6.2 d) that the functions Ξ i (f k ) are bounded on [, c s), with a bound that does not depend on k or i. Thus we have for every i IN that c s Ξ i (f k )(a) Ξ i (f l )(a) da, k, l.

25 EVENTUAL COMPACTNESS AND AGE-STRUCTURED MODELS 719 Now T (s)f k T c s (s)f l = Hence, for any i IN, c s c s + + c s lim k,l T (s)f k T (s)f l lim U(s + a, a)f k (a) U(s + a, a)f l (a) da U(s + a, a)f k (a) Ξ i (f k )(a) da Ξ i (f k )(a) Ξ i (f l )(a) da Ξ i (f l )(a) U(s + a, a)f l (a) da. c s sup k c s U(s + a, a)f k (a) Ξ i (f k )(a) da + lim sup U(s + a, a)f l (a) Ξ i (f l )(a) da. l (6.3) By definition of Ξ i (f k ) and Assumption 6.2 d), there exists some constant Λ >, such that Λ = Λ c s c s 1 U(s + a, a)f k (a) Ξ i (f k )(a) da ( 1 U(s + a, a)f k (a) U(s + a + r i, a + r i )f k(a + r ) i ) dr da ( c s U(s + a, a)f k (a) U(s + a + r i, a + r i )f k(a + r ) i ) da dr. By iii), c s U(s + a, a)f k (a) Ξ i (f k )(a) da, as i, uniformly in k IN. Taking the limit i in (6.3), we have lim sup T (s)f k T (s)f l =. k,l We now prove some results that will be used to verify Assumptions 5.4 d)-2) and d)-3. Proposition 6.5. Let Ω be a set and µ a non-negative measure on a σ-algebra of measurable subsets of Ω. Further let Z, Y, Z be normed vector spaces, K : Z Y a compact linear operator, and V a bounded linear operator from Y to L (Ω, Z ). Assume that, for every y Y, there exists a subset N y of Ω such that µ(n y ) = and (V y)(ω \ N y ) has compact closure in X. Then there exists a bounded measurable map W : Ω L(Z, Z ) and a measurable subset N of Ω such that µ(n) = and W (a)z = (V Kz)(a) for all z Z and all a Ω \ N. Further, for every ɛ >, there exist n N and measurable sets Ω 1,..., Ω n Ω and elements a 1 Ω 1,..., a n Ω n such that Ω = n Ω j N and j=1 W (a) W (a j ) < ɛ a Ω j, j = 1,..., n. We write for a disjoint union.

26 72 P. MAGAL AND H.R. THIEME Proof. Let l IN. Since K is a compact operator, there exist m l IN and z 1,..., z ml Z with z k 1 such that, for all z Z with z 1, there exists some k {1,..., m l } such that Kz K z k < 1 l+1. By assumption, for every k = 1,..., m l, there exists a measurable set N l,k in Ω such that [V Kz k ](Ω \ N l,k ) has compact closure in Z. Set N l = m l k=1 N l,k, N = l IN N l. Then µ(n) = and [V Kz k ](Ω \ N) has compact closure for all k = 1,..., m l. Let us fix l for a moment and set m = m l. We define W (a) = ( V Kz 1 (a),..., V Kz m (a) ), a Ω \ N. Then W : Ω Z m, where Z m is the m-fold Cartesian product of Z with itself, endowed with the maximum norm. Further W (Ω \ N) [V Kz 1 ](Ω \ N)... [V Kz m ](Ω \ N). The latter set has compact closure as the Cartesian product of sets with compact closure. Hence W (Ω \ N) has compact closure itself. Thus there exist n IN and elements a 1,..., a n Ω \ N such that for every a Ω \ N there exists some j {1,..., n} with W (a) W (a j ) < 1 l+1, i.e. [V Kzk ](a) [V Kz k ](a j ) 1 < k = 1,..., m. l + 1 We set m Ω j = {a Ω \ N; [V Kzk ](a) [V Kz k ](a j ) 1 < l + 1 }. k=1 Then Ω = n Ω j=1 j and a j Ω j. Since V Kz k L (Ω, Z ), Ω j is a measurable set. Now let a Ω j, j {1,..., n}, z Z, z 1. We choose k {1,..., m} such that Kz Kz k < 1 l+1. Then [V Kz](a) [V Kz](a j ) [V Kz](a) [V Kz k ](a) + [V Kz k (a) [V Kz k ](a j ) + [V Kz k ](a j ) [V Kz k ](a) <2 V Kz Kz k + 1 l + 1 < 2 V + 1. l + 1 We set W (a)z = [V Kz](a) for a Ω \ N and W (a) = for a N. Notice that this definition does not depend on l. For every j = 1,..., n, W (a) W (a j ) < (2 V + 1) 1 l+1 for all a Ω j. We set Ω 1 = Ω 1 \ {a i } and j Ω j+1 = Ω j+1 \ Ω j i>1 {a i }, j = 1,..., n 1. i=1 i>j+1 Then Ω = n j=1 Ω j N, a j Ω j, for j = 1,..., n, (since a j Ω j for j = 1,..., n), and W (a) W (a j ) < 2 V + 1 a Ω j. l + 1 W : Ω L(Z, Z ) is measurable because for every l IN we find a finite-valued measurable function W l : Ω L(Z, Z ), W l (a) = W (a j ) for all a Ω j, W l (a) = for all a N, such that W (a) W l (a) 2 V + 1 l + 1 a Ω.