Age-dependent equations with non-linear diffusion

We consider the well-posedness of models involving age structure and non-linear diffusion. Such problems arise in the study of population dynamics. It is shown how diffusion and age boundary conditions can be treated that depend non-linearly and possibly non-locally on the density itself. The abstract approach is depicted with examples.


INTRODUCTION
We consider abstract non-linear problems that naturally arise in the study of the dynamics of populations structured by age and spatial position (e.g. see [19] and the references therein). More precisely, we are interested in Banach-space-valued solutions to equations of the form The function u = u(t, a) usually represents the population density of a certain specie at time t > 0 and age a > 0, so thatū(t) in equation (1.4) is the (weighted) total population independent of age. The operator  4) are the non-linear dependence of the operators A and B on the (total) density u. While a great part of the research so far focused on linear diffusion, it is the aim of this paper to present an approach in an abstract setting giving a framework for a larger class of problems of the form (1.1)- (1.4). This will not only provide us with some flexibility in choosing the underlying functional spaces in concrete applications, but also allows us to consider non-linear diffusion and ageboundary conditions that may depend locally or possibly non-locally with respect to time on the density u. The approach applies to general second order time-dependent elliptic operators on a smooth domain Ω ⊂ R n , e.g. to operators of the form with some suitable birth modulus b (e.g., see [19]), or also age boundary conditions with history-dependent birth function of the form B[u](t) = ∞ 0 b t, a, 0 −τū (t + σ)dσ u(t, a) da as contemplated in [7], where τ > 0 is the maximal delay. We refer to our examples in Section 5.
In the next section, Section 2, we first list our assumptions and introduce the notion of a (generalized) solution to (1.1)-(1.4) before stating our main results on the well-posedness of (1.1)- (1.4). This section is then supplemented with further properties of the solution such as regularity, positivity, and global existence. The proof of the main result, Theorem 2.2, will be performed in Section 3, while the proofs of the additional properties will be given in Section 4. Finally, in Section 5 we briefly indicate how to apply these results in problems occurring in different situations of population dynamics.
We shall point out that other notions of solutions and other solution methods for age structured equations with linear diffusion were also introduced in literature, e.g. using integrated semigroups (see [11,12] and the references therein) or using perturbation arguments (see [13,14,15]). For a similar approach as in the present paper we refer to [8,9,17,18,19]. We also refer to [4,5,6,10] for other approaches to age structured equations with non-linear diffusion.

MAIN RESULTS
In the following, we assume that E 1 and E 0 are Banach spaces such that E 1 is densely and continuously embedded in E 0 . Furthermore, (·, ·) θ is for each θ ∈ (0, 1) an admissible interpolation functor, that is, E 1 is densely embedded in each E θ := (E 0 , E 1 ) θ . Let L(E 1 , E 0 ) denote the space of all bounded and linear operators from E 1 into E 0 equipped with the usual uniform operator norm. Given ω > 0 and κ ≥ 1 we write provided A ∈ L(E 1 , E 0 ) is such that ω + A is an isomorphism from E 1 onto E 0 and satisfies We set which (equipped with the topology induced by the uniform operator norm) is an open subset of L(E 1 , E 0 ). It is well known that A ∈ H(E 1 , E 0 ) if and only if −A, considered as a linear operator in E 0 with domain E 1 , is the generator of a strongly continuous analytic semigroup on E 0 , e.g. see [1]. Next, we fix a function g ∈ L + ∞,loc (R + ) satisfying for some numbers g j > 0, and we introduce the Banach space If T > 0, we put I T := [0, T ]. Given a function u ∈ E IT 0 , we simply write u(t, a) for t ∈ I T and a > 0 instead of u(t)(a). For an interval J we setJ := J \ {0}.
Throughout we suppose that there exists a number α ∈ [0, 1) such that the following assumptions hold: g(a) < ∞, and there exists ζ > 0 such that for each (A 2 ) Given T 0 , R > 0 and θ ∈ (0, 1) there are numbers ρ ∈ (0, 1), ω > 0, κ ≥ 1, σ ∈ R, and c 0 > 0 (depending possibly on θ, T 0 , and R) such that for each T ∈ (0, and for t ∈ I T , a > 0, and ū Eα , ū * Eα ≤ R. The latter assumptions in (A 2 ) and (A 3 ) guarantee that equations (1.1)-(1.4) pose a proper time evolution problem, that is, the solution depends at each time t only on the past but not on the future. In Section 5 we will give concrete examples for operators A and B satisfying (A 2 ) and (A 3 ), respectively. In particular, it will be shown that if A depends locally with respect to time onū and if E 1 is compactly embedded in E 0 , then (A 2 ) is rather easy to verify in applications (see Proposition 5.1 and Corollary 5.2). Introducing the function g in the definition of the spaces E θ allows to give a meaning to (1.4) for u ∈ E IT 0 in view of assumption (A 1 ). Also note that (2.5) is trivially satisfied if m is non-negative or bounded.
In order to introduce the notion of a solution to (1.1)-(1.4), we first observe that ifū : for t ∈ J and a > 0, where mū(t, a) := m t, a,ū(t) .
The notion of a generalized solution is derived by integrating (1.1)-(1.4) formally along characteristics. Proposition 2.5 below gives more details regarding further regularity of generalized solutions.
We first state an existence and uniqueness result for generalized solutions to (1.1)-(1.4).
A proof of this theorem will be given in Section 3. Before providing more properties of the generalized solution, we shall emphasize that the regularity assumptions on the operators A and B in (A 2 ) and (A 3 ) are imposed to overcome the difficulties induced by the quasi-linear structure of A = A[ū]. Indeed, in the case of "linear diffusion", that is, if A = A(t) depends possibly on time but is independent ofū, less assumptions are required. For simplicity, we state the following remark for a function m = m(t, a) that is independent ofū. Remark 2.3. Suppose that A ∈ C ρ (R + , H(E 1 , E 0 )) for some ρ > 0, and for each T > 0 let there be numbers 0 ≤ α ≤ β ≤ 1 with (α, β) = (0, 1) such that the function B : C(I T , E β ) → C(I T , E α ) is uniformly Lipschitz continuous on bounded sets and satisfies B[u]| IT = B[u * ]| IT for 0 < T < S, u, u * ∈ C(I S , E β ), and u| IT = u * | IT . If m ∈ C(R + × R + ) is bounded, then the problem admits for each u 0 ∈ E β a unique maximal generalized solution u ∈ C(J, E β ), which exists globally if (2.7) holds.
A proof of this remark follows along the lines of the proof of Theorem 2.2 and we thus omit details.
We now give additional properties of the generalized solution. For the rest of this section, we suppose the assumptions of Theorem 2.2, and we fix u 0 ∈ E β and let u = u(·; u 0 ) ∈ C(J, for T ∈J and β ≤ υ ≤ 1 denote the unique maximal generalized solution to (1.1)-(1.4) on J = J(u 0 ) corresponding to u 0 .
First we mention that the solution depends continuously on the initial value u 0 ∈ E β . More precisely, we have Next, we note that the solution possesses more regularity if the date are more regular.

Proposition 2.5. Suppose that
In addition, let Then, for all t ∈J and a > 0, we have Since u represents a density in applications, one expects it to be non-negative. The next result establishes this positivity result if E 0 is an ordered B-space with positive cone E + 0 . In this case we put We refer to [1] for more information about operators on ordered B-spaces.

13)
and where Further suppose that m a non-negative or bounded. Then the solution u exists globally, that is, J = R + .
is a consequence of (2.4). Also, condition (2.14) may be replaced by for (0, 1) = (ϑ, υ) ∈ [0, 1] 2 and some c 2 = c 2 (T ) > 0. This latter condition is slightly weaker than (2.14) with respect to regularity since we may allow for υ > β, but it somehow assumes B to depend locally on u with respect to time.
For the proofs of Corollary 2.4, Propositions 2.5-2.7, and Remark 2.8 we refer to Section 4.

PROOF OF THEOREM 2.2
Given the assumptions of Theorem 2.2, let γ ∈ [α, β) be arbitrary and choose θ ∈ (0, ζ ∧ (β − γ)). We fix any T 0 > 0 and R with and observe that V T , equipped with the topology induced by C(I T , E γ ), is a complete metric space. Also note that (A 1 ) ensures the existence of a constant c 1 > 0 such that and hence and thus there are numbers ρ ∈ (0, 1), ω > 0, κ ≥ 1, σ ∈ R, and c 0 > 0 depending on T 0 and R such that (2.2), (2.3) hold for u, u * ∈ V T . Therefore, invoking Lemma II.5.1.3, Lemma II.5.1.4, and Equation (II.5.3.8) in [1], we conclude that there exists c(T 0 , R) > 0 such that unique evolution systems for 0 ≤ r < s < t ≤ T and 0 < υ ≤ τ < 1, as well as for 0 ≤ t ≤ T , a > 0, and u ∈ V T , we claim that Θ : is chosen sufficiently small. In the following, letμ ∈ (0, µ). We first prove that Θ(u) ∈ C(I T , E β ) ֒→ C(I T , E γ ) for u ∈ V T . To this end, observe that assumptions Hence, recalling that g ∈ L ∞,loc (R + ) and using (2.5), (3.5), (3.6), (3.9), and (3.11) we estimate for u Now, as |t − t * | → 0 we clearly have I + IV + V + V I → 0 due the Lebesgue Theorem (we obviously may assume β ≥ µ). Using B[u] ∈ C(I T , E µ ), the density of the embedding E β ֒→ E µ , and the fact that the evolution system U A[ū] is uniformly strongly continuous on compact subsets of E β , we also derive that II → 0. The continuity of B[u] also entails III → 0, while the strong continuity of U A[ū] on E β ensures V II → 0. Finally, V III → 0 holds since translations are strongly continuous. Therefore, Θ(u) ∈ C(I T , E β ). Next observe that (3.7) implies where In view of (3.1), (3.5), (3.9), (3.11), (3.12), and assumption (A 4 ) we deduce, for u ∈ V T and t ∈ I T , that Since γ < β we may choose T = T (R) ∈ (0, T 0 ) sufficiently small to obtain Moreover, writing for u ∈ V T and t ∈ I T and using the fact that, for 0 ≤ a ≤ t ≤ t * ≤ T , Taking into account that, due to (A 1 ), and recalling the choice of θ, we may make T = T (R) ∈ (0, T 0 ) smaller, if necessary, and conclude that To prove that Θ is contractive, we observe that assumption (A 4 ) together with (3.1), (3.5), (3.7), (3.8), (3.9), and (3.10) imply that, for u, u * ∈ V T , 0 ≤ t ≤ T ≤ T 0 , and for all ξ ∈ [0, β], In particular, taking ξ = γ < β we may choose T = T (R) ∈ (0, T 0 ) sufficiently small such that Therefore, Θ : V T → V T is a contraction provided T = T (R) ∈ (0, T 0 ) is sufficiently small and hence possesses a unique fixed point, say u, in V T ∩ C(I T , E β ). Consequently, for 0 ≤ t ≤ T and a > 0. An estimate similar to (3.13) combined with the strong continuity properties of the evolution system U A[ū] then warrants that In order to extend the just found solution u ∈ C(I T , E β ), we choose similarly as in (3.1), and take now V S to be and obtain V ∈ C(I T +S , E γ ) with V C θ (IT +S ,Eγ ) ≤ R + 2. We then introducê for t ∈ I S and a > 0. It follows from assumption (A 2 ) that for some σ, ω, κ, ρ depending on R and T 0 . If also v * ∈ V S , then HenceÂ satisfies (2.2) and (2.3). Moreover, for v, v * ∈ V S we also have , that is,B satisfies (2.4). Taking S = S(R) > 0 sufficiently small we deduce as before the existence of a function v ∈ C(I S , for 0 ≤ t ≤ S and a > 0. We then extend the function u by w : I T +S → E β being defined as Hence the function w still satisfies (3.17) in which u is replaced by w everywhere. Next, sincê we have by uniqueness

From these observations and using (3.17) and (3.20) it is then straightforward that
for T ≤ t ≤ T + S. Therefore, we may extend u to a unique maximal generalized solution u in C(J, E β ) satisfying The right hand side is clearly differentiable with respect to t and, owing to (3.17) and (3.18)

Proof of Proposition 2.5. To establish Proposition 2.5 we use the properties of evolution systems [1]
Then, due to (2.8) and (2.9), it follows from (3.17) that, for t ∈J and a > 0 with a = t, if m ∈ C 0,1 (J × R + ) and similarly t * ≤ T , and using the fact that the resolvent positivity of the operator A implies [1,II.6.4], it follows from the assumptions on B that the map Θ, introduced in the proof of Theorem 2.2, is a contraction from V + T into itself (provided T is chosen sufficiently small). This then readily gives Proposition 2.6.

APPLICATIONS
We give examples of problems to which the results of Section 2 may be applied. First we provide some conditions intended to simplify the verification of assumption (A 2 ) and (2.12), (2.13).

General Remarks.
We show that if A has a particular form, then assumption (A 2 ) is rather easy to verify in concrete applications. This result, in particular, applies to the case when A depends locally with respect to time onū.
More precisely, we assume that A is of the form where for some Banach space F 0 and ̺ ∈ (0, 1). That is, given any R > 0 there exists c(R) > 0 such that for t, t * ∈ [0, R] and z, z * ∈ F 0 with z F0 , z * F0 ≤ R. Given another Banach space F 1 with F 1 ֒→ F 0 , the function Φ is supposed to satisfy the following conditions (for some α ∈ [0, 1)): (A 5 ) Given T 0 , R > 0 and θ ∈ (0, 1) there are numbers ρ ∈ (0, 1) and c 4 > 0 (depending on T 0 , R, and θ) such that, for each T ∈ (0, T 0 ), the function Φ maps C θ (I T , E α ) into C ρ (I T , F 1 ) and satisfies for t, t * ∈ I T and allz,z Then we have: Proof. Given T 0 , R > 0 and θ ∈ (0, 1) it follows from assumption (A 5 ) that there exists a bounded set Due to the compactness of the embedding F 1 ֒→ F 0 we deduce that M is relatively compact in F 0 and so It is worthwhile to point out that assumption (A 5 ) is trivially satisfied if Φ is the identity. Therefore, assumption (A 2 ) holds for operators A depending locally with respect to time onz:
If A is of the form (5.1), then also the conditions (2.12), (2.13) for global existence are simpler to verify. Thus we consider again the unique maximal generalized solution u = u(·; u 0 ) ∈ C(J, E β ) to (1.1)-(1.4) on J = J(u 0 ) corresponding to u 0 as provided by Theorem 2.2.

5.2.2.
A tumor invasion model. The following haptotaxis model describes the invasion of tumor cells (with density u) into the surrounding tissue along gradients of bound cell adhesion molecules (with density f ) that are contained in the extracellular matrix. The cells produce a matrix degradative enzyme with density v. The model was studied in detail in [17,18], and we just recall a very simple version: (t, a, x) ∈ R + × R + × Ω , (5.22) If χ is smooth and D satisfies (5.5), we obtain for there exists a unique non-negative solution (f, v, u) ∈ C(R + , X) to (5.22)-(5.28), f and v being classical solutions to the corresponding equations. Moreover,ū ∈ C 1 (R + , L p ) ∩ C(R + , W 2 p,B ).

5.2.3.
Swarm-colony development of Proteus mirabilis. Finally, we mention another example that fits into the abstract framework of (1.1)-(1.4). The model describes the swarming phenomenon of a bacterium called Proteus mirabilis. It models the evolution of mononuclear "swimmers" with density v and multi-cellular