PARABOLIC REACTION-DIFFUSION NONLOCAL COUPLED DIFFUSIVITY TERMS

. In this work we study a system of parabolic reaction-di(cid:27)usion equations which are coupled not only through the reaction terms but also by way of nonlocal di(cid:27)usivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we (cid:28)nd the conditions under which the existence of strong solutions is assured. We establish several bow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.


Introduction
In this work, we are interested in studying systems of reaction-diusion equations of the form where τ = 0 or τ = 1, and n denotes the outward unit normal to ∂Ω. Throughout this work, we will consider either the case of Dirichlet boundary conditions, i.e. τ = 0 in (1.4)-(1.5), or of Neumann boundary conditions, i.e. τ = 1 in (1.4)- (1.5). In the nal part, we will distinguish situations where it is important to consider specic boundary conditions. Note that the existence of the unit normal to ∂Ω in almost all points of ∂Ω implies that ∂Ω is suciently regular, for instance Lipschitz-continuous. E-mail address: ferreirajorge2012@gmail.com, CORRESPONDING AUTHOR: holivei@ualg.pt.
Systems of reaction-diusion equations are very important in the applied sciences to model interesting and very distinct phenomena, where many chemical and biological processes are in the rst line of its applications. On the other hand, the combination of the coupling diusions together with the coupling reactions produces many mathematical features. In particular, systems of reaction-diusion equations lead to the possibility of many threshold phenomena that we cannot expect they happen if we consider only one reaction-diusion equation. An interesting feature of the reaction-diusion equations (1.1)-(1.2) observed in many models, arise when the diusion coecient, say for simplicity p(u), is given by a local quantity. However, in many applications this assumption is incompatible with the physical notion of measure, since we are not able to measure pointwisely the diusivity of a pointwise density. One possibility to overcome this diculty, consists in choosing a point x in the space and then constructing a ball B := B(x, ) centered at x with radius and replacing p(u) by (1.6) p − x (see e.g. [18]). On the other hand, systems of reaction-diusion equations, but with nonlocal reaction terms instead of diusion ones, were recently proposed to describe the motion of particle densities under the presence of some chemical reactions (see [16]) and to model the evolution of a population under chemotactic eects (see [19]). Though our motivation to study the system (1.1)-(1.3) is primarily mathematical, we can nd some interesting aspects of its applications in population dynamics. See, for instance, the references [3,8,17,18].
During the last decades a lot of attention has been devoted to nonlocal diusion and reaction-diusion problems. In [4,18], the existence and uniqueness of local and global solutions to the following parabolic diusion problem has been proved. Here, the diusivity a is some function from R into (0, +∞) and l is a continuous mapping from L 2 (Ω) into R. The authors have worked on dierent problems for distinct diusivity terms, but always depending on´Ω u dx, and under dierent boundary conditions: Dirichlet, Neumann and mixed boundary conditions. In [5], besides proving the existence and uniqueness of solutions, the same authors have analyzed the asymptotic behavior of the solutions as well. These issues were extensively investigated in [7] where, in particular, the convergence of the solutions to a steady state was proved. Several extensions and modications of the problem (1.8) were deeply studied in [2], where many interesting examples were given as well. Again the authors of [4,5,18], considered, in [6], a class of nonlocal elliptic and parabolic problems related to (1.8), now with homogeneous Dirichlet boundary conditions, for which they proved existence and uniqueness results. The analysis of the problem (1.8), considered with a nonlocal diusivity depending on the Dirichlet integral´Ω |∇u| p dx, was carried out in [10,23] for p = 2 and in [9] for a general p (and for the p-Laplacian). The asymptotic behavior of the solutions to the problem (1.8), considered with a nonlocal diusivity written as a kernel, i.e. l(u) =´Ω g u dx, where g is a given function in L 2 (Ω), has been performed in [23] too.
Reaction-diusion analogues of the parabolic problem (1.8) were considered by the authors of [1,13] in the following form In [1], the problem (1.9) was considered in a rather general Banach space and the authors worked on the case a(u) = ´Ω u dx −1 . This assumption led them to an equivalent reaction-diusion problem with a nonlocal diusivity, now multiplied by the reaction term f (u). For these problems, the authors established local existence and uniqueness results and, in addition, they found conditions on the initial data in order to obtain time properties of nite extinction or persistency of the solutions. In [13], the authors extended the results of [4,5,6,7,10,18] to the case of the reaction-diusion problem (1.9). In particular, they considered both stationary and transient situations, where the nonlinearity appears, not only in the nonlocal diusivity term a(l(u)), but also on the right-hand side in which one has the nonlinear function f (u). The outline of our work is the following. In Section 2, we dene the notion of weak solutions to the problem (1.1)-(1.5) and we present Theorem 2.1 where is established the existence of weak solutions. The proof of Theorem 2.1 is carried out in Section 3 by using Galerkin approximations together with compactness arguments. In Section 4, we drop the boundedness condition on the nonlocal functions (see (2.2)) to prove a local existence result in Theorem 4.1. Section 5 is devoted to prove the uniqueness result and in Section 6 we nd the conditions under which we prove the existence of strong solutions. In Section 7, we establish several bow-up results for the strong solutions to the problem (1.1)-(1.5). Finally, in Section 8, we give a criterium for the convergence of these strong solutions towards a homogeneous state by using the theory of invariant regions.
The notation used throughout this work is largely standard in the eld of Partial Dierential Equations and we address the reader to the monographs [3,15,20,22] for any question related with this matter.
In any case, V τ is a closed subspace of H 1 (Ω), with its norm satisfying to for some positive constants C 1 and C 2 . Denition 2.1. Let N ≥ 2 and assume that conditions (2.1)-(2.6) hold. We say (u, v) is a weak solution to the problem (1.1)-(1.5), for either τ = 0 or τ = 1, if: In order to prove the existence of weak solutions to the problem (1.1)-(1.5), we have to impose a suitable restriction related with the Poincaré inequality. We assume that the constants of uniform parabolicity m i and of Lipschitz continuity C Li are related by where λ P is the principal (positive) eigenvalue for the Laplacian problem Observe that, in the case of Neumann boundary conditions, i.e. τ = 1 in (2.10), 0 is clearly an eigenvalue, with the associated eigenfunction given by any constant, which in turn can be xed by a normalization such as φ = 0, where φ = − Ω φ dx. In any case, the Rayleigh quotient allows one to characterize the principal (positive) eigenvalue of (2.10) with the following minimum principle, .
3. Proof of Theorem 2.1 3.1. Existence of approximative solutions. In order to use the Galerkin method, let {φ i } ∞ i=1 be a set of non-trivial solutions φ i , associated to the eigenvalues λ i > 0, i = 1, 2, ..., to the following spectral problem: is orthogonal in V τ and can be chosen as being orthonormal in L 2 (Ω) (see e.g. [15]).
Given m ∈ N, let us consider the correspondingly m-dimensional space V m τ spanned by φ 1 , φ 2 , ..., φ m . For each m ∈ N, we search for an approximative solution (u m (t), v m (t)) of (2.7)-(2.8) in the form where φ k ∈ V m τ are given and c m k (t) and d m k (t) are the functions we look for. These functions are found by solving the following system of 2m nonlinear ordinary dierential equations, with respect to the 2m where both u m 0 and v m 0 are chosen in such a way that Attending to the continuity of a 1 , a 2 and f , g on u and v (see 3) by d m k , where in both it is taken ϕ = ϕ k and η = ϕ k , we add up the resulting equations from k = 1 until k = m and then we integrate them between 0 and t, with t ∈ (0, t m ), to obtain (3.8) Then we use Cauchy's inequality together with the Poincaré inequalities (2.12)-(2.13) on the fth and sixth terms of (3.8), which yield Observe that, by the denition of u m (t), v m (t) and of V m τ set forth in (3.1), we can use (2.13) with − Ω u m (t)dx = 0 and − Ω v m (t)dx = 0. Now we use the information of (3.9)-(3.10) in (3.8) which, in view of (3.5), yields Finally, assumption (2.9) guaranties that On the other hand, by using the equations (3.2)-(3.3) together with (3.11) and with the assumptions (2.2), (2.3)-(2.4) and (2.5)-(2.6) and still using the Poincaré inequalities (2.12)-(2.13), it can be proved the existence of positive constants C 1 = C(M 1 , C 0 ) and C 2 = C(M 2 , C 0 ), where C 0 is the constant from the inequality (3.11), such that Hence, by means of reexivity, Now, due to (3.13) and (3.15), and observing the compact and continuous imbeddings V τ → → L 2 (Ω) → V τ hold, we can use Aubin-Lions compactness lemma to prove that Thus, from the assumptions (2.3)-(2.4) and (2.5)-(2.6), we have On the other hand, from the continuity of p, q, r and s (see (1.7)) and from the continuity of a 1 and a 2 (2.1), we can use (3.16) to prove that Then, from Riesz-Fischer theorem we have, up to some subsequences,  .2) and (3.3) to the limit m → ∞ to prove that (2.7) and (2.8) hold in D (0, T ), rst for all ϕ, η ∈ {φ 1 , . . . , φ m }, then, by linearity, for all ϕ, η ∈ V m τ and next, by continuity, for all ϕ, η ∈ V τ . In particular, and once that by (3.13) Then, arguing as we did for (3.11), but taking the supreme, we obtain from (3.23)-(3.24) that On the other hand, observe that we can writê Using (3.4)-(3.5) and (3.15), we can pass the above equation to the limit m → ∞ to obtain Consequently u(0) = u 0 . By a completely analogous reasoning, we also have v(0) = v 0 . The proof of Theorem 2.1 is thus concluded.

Local existence
In this section, we establish a local version of Theorem 2.1. This result shall be proved under the assumptions that the nonlocal functions a 1 and a 2 are strictly positive in some neighborhoods. Before we establish the existence result of this section, let us x some notation rst. For each i ∈ {1, 2}, we consider the open ball B δi (ξ i , η i ) and the closed ball B δi (ξ i , η i ) centered at (ξ i , η i ) ∈ R 2 and with radius δ i . We stress here that the functions a 1 and a 2 have the arguments satisfying to (1.7). Theorem 4.1. Assume that all the conditions of Theorem 2.1 are satised, with the exception of (2.2). In addition, assume that for some (ξ 1 , η 1 ), (ξ 2 , η 2 ) ∈ R 2 and for some δ 1 , δ 2 > 0. If then there exists T 0 > 0, and a weak solution (u, v) In the proof of this result, we follow the approach of [5].

Uniqueness
Here, we will adapt the results of [8,13] to establish an uniqueness result. Lipschitz conditions on the nonlocal diusivity terms and on the reaction functions (already assumed at (2.3) and (2.5)) play a fundamental role.

Strong solutions
In this section we will nd conditions on the data of the problem (1.1)-(1.5) under which the solutions found in the previous sections are more regular. We prove that the time derivatives u t and v t are square sumable in Q T and we establish a result that gives us more spatial regularity for the solution (u, v).
Here, and in addition to (3.4)-(3.5), we assume these approximations satisfy to (6.5) ∇u m (0) = D m 0 , where D m 0 is chosen in such a way that (6.6) D m 0 → ∇u 0 strongly in L 2 (Ω), as m → ∞. Next, we consider a sequence of functions n ∈ C 1 (0, T ), with n ∈ N, such that for any n ∈ N. We take 2) and we add up the resulting equation from k = 1 until k = m. Hence, we obtain where we have used integration by parts on the second term of (6.7) together with (6.5). The assumption that a 1 ∈ C 1 (0, T ) (see (6.2)) and the Cauchy-Schwarz inequality (on the last term) were also used in the derivation of (6.8). Then, using the assumptions (2.2) and (2.3) together with the properties of the sequence n , and the fact that a 1 is non-increasing in t (see (6.2)), we obtain 1 2ˆT Letting n → ∞ rst and then making m → ∞, we obtain where we have used (3.16) and (6.6). Finally, due to (3.13) and to (6.1), we conclude that u t ∈ L 2 (0, T ; L 2 (Ω)). Analogously, it can be proved that v t ∈ L 2 (0, T ; L 2 (Ω)).
The next step is to prove (6.4). To prove this, let us consider a xed, but arbitrary, open bounded domain U ⊂⊂ Ω and let us choose another open bounded domain W such that U ⊂⊂ W ⊂⊂ Ω. Then we consider a function ζ ∈ C ∞ (R N ) such that We consider the dierence quotient D h k u(t) of the (weak) partial derivative u(t), dened by for x ∈ U and h ∈ R \ {0} such that |h| < dist(U, ∂Ω). Then we take for test function in (2.7) a.e. in t . We observe that whenever the following relations are possible, we have for all admissible functions θ and ϑ (see e.g. [15] Now, using the assumptions (2.2) and (2.3) together with the Cauchy-Schwarz inequality, and observing that 0 ≤ ζ ≤ 1 and |∇ζ| ≤ C, where C is a positive constant, we have (6.10) Then, observe that, by the properties of the dierence quotients (see e.g. [15]), there exists a constant C 0 such thatˆΩ Gathering the information of (6.10)-(6.11), choosing an such that 0 < < m1 C0+M1 and using the reasoning of (6.11) on the last term of (6.10), we get 12) where the positive constant C depends on m 1 , C L1 and C 0 . Using a well-known result of the dierence quotients (see e.g. [15,Theorem 5.8.3]), we obtain for a.a. t ∈ [0, T ] Integrating the last relation in the interval [0, T ] and using (3.13) and (6.3), we prove nally that u ∈ L 2 (0, T ; H 2 (Ω)). Analogously, it can be proved that v ∈ L 2 (0, T ; H 2 (Ω)).

Existence of Blow-up
In this section, we will establish several bow-up results for the strong solutions to the problem (1.1)-(1.5). By a strong solution, we mean here a solution (u, v) in the conditions of Theorem 6.1. For a given solution If t * = ∞, the solution (u, v) is global, since, as in the local problem, it can be shown (see e.g. [14]) that u and v can be continued for all times t > 0. On the other hand, if t * < ∞, we have When this happens, we say the solution (u, v) under consideration blows up in the nite time t * . Blow-up criteria for systems of parabolic equations are normally more dicult to nd than for the scalar case. The following version of Jensen's inequality will allow us to develop some blow-up criteria to our reaction-diusion system (1.1)-(1.5). Lemma 7.1. Let Ω ⊂ R N be a bounded domain and assume that F : R 2 −→ R is convex. Then for every u, v ∈ L 1 (Ω), Proof. Due to the convexity of F , for each (x 1 , x 2 ) ∈ R 2 there exists (z 1 , z 2 ) ∈ R 2 such that holds for all (y 1 , y 2 ) ∈ R 2 , i.e. the graph of F lies above its supporting hyperplane at (x 1 , x 2 ). In this inequality, let us take x 1 = − Ω u dx, x 2 = − Ω v dx, y 1 = u and y 2 = v. This yields Then, integrating over Ω, with respect to x, and observing that the terms which are multiplied by z 1 and z 2 vanish, we immediately arrive at (7.2).
As a rst example of the utility of the Lemma 7.1, we have the following blow-up result under Neumann boundary conditions. Theorem 7.1. Let (u, v) be a couple of strong solutions to the reaction-diusion system (1.1)-(1.5) endowed with the Neumann boundary conditions, i.e. with τ = 1 in (1.4)-(1.5). Assume that (1) f and g are convex functions, (2) then the solution (u, v) to the reaction-diusion system (1.1)-(1.5) with τ = 1 blows-up in the nite time t * . Remark 7.1. Some functions satisfying to condition (2) are, in the case of u 0 + v 0 > 0, f (s, r) = |r| p and g(s, r) = |s| q for suitable p ≥ q > 1, or still more general f (s, r) = a|r| p + b|s| q and g(s, r) = c|r| p + d|s| q , where a, b, c, d are positive real constants and p, q, p, q > 1. In the case of u 0 + v 0 ≤ 0, one should consider, for instance, examples of the form f (u, v) = g(u, v) = h(u + v) = (1 + |u + v|) p , for some p > 1, Proof. Adding up the equations (1.1) and (1.2), we obtain Let use the notations u(t) = − Ω u(t) dx and v(t) = − Ω v(t) dx. Integrating the above equation over Ω, using Gauss-Green's theorem together with (1.4)-(1.5) with τ = 1, and invoking the nonlocal character of a 1 and a 2 , we obtain Then, Lemma 7.1 and assumption (2) yield Finally, integrating between 0 and t > 0 and using (1.3) together with (7.3), we obtain Then, from a well-known result (see e.g. [3,Theorem 13.11]), we conclude that u + v will blow up in the nite time t * provided that h(w) > 0 for all w ≥ u 0 + v 0 . That (u, v) blows up in the sense of (7.1), is an immediate consequence.
then the rst component of the solution (u, v) to the reaction-diusion system (1.1)-(1.5), with τ = 0, blows-up in the nite time t * . Proof. We start by multiplying the equation (1.1) by ϕ, we integrate over Ω and we use (1.1), with τ = 0, and (2.10) together with the nonlocal character of a 1 . After all, we obtain Observing (7.4), we can use Jensen's inequality (7.2), to prove that Replacing this into the previous equation and, in addition, using the hypothesis that It should be noted that µ(t) is well dened on the existence interval of the solution u. Then, integrating the last inequality between 0 and t > 0, and using the fact that µ(0) ≥ 0 and hypothesis (7.5), we obtain Then, from [3, Theorem 13.11], we conclude that µ(t), and consequently u, will blow up in a nite time provided that f (w, 0) + λ P a 1 w > 0 for all w ≥ µ(0).
Remark 7.2. We observe that according to the proof of the last result, we had no need to use the boundary condition v = 0 on ∂Ω. Therefore, we still have blow up of the rst component even if v is not prescribed at the boundary.
We end this section by giving a criterium of blow up of both components of the solution (u, v) to the reaction-diusion system (1.1)-(1.5) endowed with Neumann boundary conditions. It should be remarked that, in the case of condition (7.1) is satised, there is, a priori, no reason for both components of the system (1.1)-(1.5) to blow up. Indeed it may happen that one of the components of (u, v) blows up as t → t − * , while the other remains bounded on [0, t − * ). Thus condition (7.1) only implies that (7.6) lim sup for possibly distinct times t 1 * and t 2 * , we shall say that both u and v blow up in nite times. When this happens at the same time t * , i.e. when t * = t 1 * = t 2 * , we say that u and v blow simultaneously (in the nite time t * ). Theorem 7.3. Let (u, v) be a couple of strong solutions to the reaction-diusion system (1.1)-(1.5) endowed with the Neumann boundary conditions, i.e. with τ = 1 in (1.4)-(1.5). Assume that (1) then both u and v blow up, one in the nite time t * and the other in another instant that can be posterior.
Proof. Arguing as we did in the rst part of the proof of Theorem 7.1, we obtain d dt Recall that u(t) = − Ω u(t) dx and v(t) = − Ω v(t) dx, and observe that in the last inequality we have again made use of the convexity of f . Then, integrating between 0 and t > 0 and using (7.7), we obtain Again, as in the proof of Theorem 7.1, we conclude that u(t)+v(t) 2 will blow up in the nite time t * provided that f (w) > 0 for all w ≥ u0+v0 2 , condition that is assured by assumption (3) and once that (7.8) min Consequently (u, v) blows up in the nite time t * and therefore (7.1) holds, which in turn only implies (7.6). Now, in order to show that both u and v blow up in nite times, we will argue by contradiction. If we assume, for instance, that v blows-up at the time t * and u does not blow up in any nite time, then we would have lim sup In the case of u 0 ≥ v 0 , then we would get, in view of (7.8), that which contradicts (7.7). For the case of u 0 ≤ v 0 , then, and again in view of (7.8), which cannot happen due to (7.7). Remark 7.3. Under the assumptions of Theorem 7.3 and for suitable reaction terms, it is possible to prove the blow-up of u and v is simultaneous. In fact, modifying the arguing of [11,21], we can prove the simultaneous blow-up of u and v in the case of reaction terms with the same shape and such that for some positive constants α and λ. An example of such a situation is the reaction function f (w) = |w| p , which satises to the above conditions for α = 1 p−1 , p > 1, and for any positive constant λ. The same reasoning can be applied to non-local problems with reaction terms similar to the ones considered in the works [11,21]. Remark 7.4. In the particular case of Theorem 7.3 with assumption (1) restricted to the case of f (u, v) = f (v) and g(u, v) = f (u), we can easily prove the simultaneous blow-up of u and v. To see this let us assume thatu and v blow-up at distinct times t 1 * and t 2 * , respectively, with t 1 * < t 2 * .

Asymptotic stability
In this section, we shall consider strong solutions (u, v) to the Neumann problem (1.1)-(1.5), in the conditions of Theorem 6.1, in a cylinder Q T0 for some T 0 > 0. The aim of the present section, is to give a criterium for the convergence of these solutions towards a homogeneous state. In particular, we will show the non-existence of nonconstant steady state solutions if certain conditions are satised. Our approach will be based on a criterium for the existence of invariant regions and then to exploit this idea to study the asymptotic behavior of the solutions (see e.g. [20,Chapter 14]). We recall that a bounded subset Σ in the uv plane is called an invariant region for a strong solution (u, v) to the problem (1.1)- (1.5) in Q T0 , if the following property is veried: (u 0 , v 0 ) and the boundary values of (u, v) on ∂Ω lie in Σ ⇒ (u, v) lies in Σ for all (x, t) ∈ Q T0 .
Observe that the assumption of h i to be quasi-convex for all i ∈ {1, . . . , m}, implies that Σ is a convex domain. In particular, a rectangular domain of the form [a, b] × [c, d] satises this condition. On the other hand, the condition (f (u, v), g(u, v)) · n < 0 on ∂Σ means that (f (u, v), g(u, v)) points to the interior of Σ on ∂Σ.
This proof can be carried over to any space dimension, though the exposition becomes too heavy.
Next we will use the notion of invariant regions to study the asymptotic behavior of the solutions to the where the reaction functions f and g are assumed to be suciently regular, and the subscript (u, v) means that the gradient of f and g is taken with respect to these variables. Observe that, by the characterization of the invariant regions set forth in (8.1), Σ is compact and therefore M < ∞. We also x the notation (8.13) σ := mλ P − 2M , where λ P is the principal eigenvalue to the Laplacian problem (2.10) with Neumann boundary conditions and m := min{m 1 , m 2 }, being m 1 and m 2 dened at (2.2).