On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion

In this article, we are concerned with the asymptotic behavior of a class of degenerate parabolic problems involving porous medium type equations, in a bounded domain, when the diffusion coefficient becomes large. We prove the upper semicontinuity of the associated global attractor as the diffusion increases to infinity.

1. Introduction. In this paper, we study the solution and associated global attractor of the nonlinear reaction-diffusion equation coupled with an initial data u 0 D ∈ L 2 (Ω) and homogeneous Neumann boundary condition, in a smooth bounded domain, as the diffusion coefficient D becomes large. We consider β an increasing continuous function in R with linear growth, g ∈ L ∞ (Ω) and the nonlinearity f satisfies certain conditions specified below. These type of equations appear in the modelling of several biological and physical problems, among them the filtration of a fluid in a partially saturated porous media (see [1]) and the evolution of a biological population (see [10] ).
The asymptotic behaviour of parabolic partial differential equations with large diffusion has been extensively studied in the literature, particularly for systems of the type ∂ t u = DLu + f (x, u, ∇u), where L is a linear operator of second-order (see [3], [4], [5] and [11]). As it is expected, from a physical point of view, since diffusion enhances spatial homogenization, the large diffusivity will cause the solutions to converge to a time dependent spatially constant function. Hence, the asymptotic behaviour is determined by an ordinary differential equation. This result has been extended to include equations that depend on a nonlinear operator, as in the case of the p-Laplace and porous medium type equations (see [16], [12] and [13]). We remark that the conditions on β and the nonlinearily f studied presently differ from the conditions imposed in [12] and [13]. Moreover, our contribution is based mainly on the study of the upper semicontinuity of the associated global attractor.
There are several studies on the existence of a global attractor for equations of porous medium type. We refer to [9] for the study of the problem in an unbounded domain. In [2] the existence of a global attractor in L ∞ (Ω) is obtained, whereas in [8] the authors proved the existence of a global attractor in L 1 (Ω).
One of the main issues when studying the long time behaviour of the equation as a dynamical system is the well posedness of the problem under relative generality of the nonlinearity f . It may provide difficulties when trying to apply the usual Galerkin method or the abstract nonlinear semigroup theory, especially to assure the uniqueness, the continuous dependence on the initial data or the necessary energy inequalities. Taking this into account, we study the problem with conditions as imposed in [6], where the existence is proved using an approximation method, studying first problem (1) with a more regular β and then passing to the limit. This approximation will be used several times to study the uppersemicontinuity of the global attractor in L 2 (Ω).
By the results in [6], we know that the Neumann boundary problem associated to equation (1) determines, for each D ≥ 1, a semigroup (S D (t)) t≥0 on L 2 (Ω) by S D (t)u 0 D = u D (·, t) where u D is the weak solution of the problem corresponding to (1). The semigroup associated with (1) admits a compact global attractor {A D } in L 2 (Ω).
Following the line of ideas in [16], we prove that the family of attractors {A D } is upper semicontinuous at infinity, which means, where A ∞ is the attractor of the limit problem defined below, for u 0 such that u 0 D → u 0 in L 2 (Ω), as D goes to infinity, and This paper is organized as follows. In Section 2, we present the definition of weak solution of the initial value problem associated to (1) and collect well known results on its well posedness and the existence of the attractor. We also recall apriori estimates uniform with respect to the diffusion parameter, which we use in Section 3, to obtain the convergence of the solutions as D goes to infinity and the problem it satisfies at the limit. In Section 4, we study the global attractor associated to the ODE obtained at the limit and finally in Section 5, we prove the upper semicontinuity of the attractor.
2. Preliminaries: Existence of solution and associated attractor. We recall in this section known results on the global existence of solutions and associated attractor, as well as apriori estimates uniform with respect to the parameter D, which we will need later to study the upper semicontinuity of the global attractor. We first give some notation that will be used throughout the paper.
Let β be a continuous increasing function with β(0) = 0. We define, for t ∈ R, We also define the sign function by We denote by W 1,2 q ((0, T ) × Ω), for all T > 0 and Ω, an open, bounded set of R N , the set of all functions v such that We review the definition of subdifferential. Let X be a Hilbert space and let Ψ : X → (−∞, ∞] be a proper lowersemicontinuous convex function. The subdifferential ∂Ψ of Ψ at u in X is defined as follows: We consider the following initial boundary value problem in a bounded smooth domain Ω ⊂ R N , N ≥ 1, with given g(x) ∈ L ∞ (Ω) and u 0 D ∈ L 2 (Ω) for each 1 ≤ D ∈ R . We assume the following conditions. (H1) β is an increasing continuous function from R into R, β(0) = 0, and there exist c 1 > 0 and c 2 > 0 such that |β(t)| ≤ c 1 |t| + c 2 for all t ∈ R. (H2) The function f is continuous and we assume there exist c 3 > 0, c 4 > 0 and c 5 > 0 such that (H3) There exists c 6 > 0 such that f (s) + c 6 β(s) is increasing almost everywhere.

Remark 1. An example is given by
By the results in [6], there exists a unique weak solution of problem (4) in the following sense.
To prove the existence of a unique solution of problem (4) in [6], the following approximation was used.
has a unique solution u j ∈ W 1,2 In the next lemma, we collect some apriori estimates which are independent of the parameter D and will be used to prove the convergence of solutions of problem (4).
Moreover, if u D is a solution of problem (4) associated to g and u 0 D , whileû D is a solution of problem (4) associated toĝ andû 0 D , then we have the following inequality We move on to the estimates on the solutions in L ∞ (Ω) and H 1 (Ω) that will allow us to conclude the existence of a global attractor, which we denote by A D .
Remark 3. The previous lemma implies the existence of an absorbing set in L q (Ω), for any q ≥ 1.
Thus, by the previous lemmas, we obtain the following result. 3. Passage to the limit. In the present section, we obtain uniform bounds in space and time, which imply the strong convergence of β(u D ) in L 1 ((0, T ]; L 1 (Ω)), as D goes to infinity. This, together with apriori estimates obtained in Lemma 2.2, provide us with the appropriate tools to obtain the problem, satisfied at the limit, when D goes to infinity.
Using the regular approximation in Proposition 1 and the arguments of Lemma 2.2 in [15] and Lemma 1.8 in [1], we have the following result.
Proof. We work with u j the solutions of the approximation problem (5) and then pass to the limit. Our first aim is to prove that there exists C > 0 such that Integrating for t ∈ (t, t + h) the equation in (5), we have

MARÍA ASTUDILLO AND MARCELO M. CAVALCANTI
and hence |E c | ≤ T − h ≤ C/M . Split the integral of β j (u j (t + h)) − β j (u j (t)) L p over (0, T − h) into an integral over E and E c . By the claim, and where C is independent of j and D, since β j (u j ) is uniformly bounded in L ∞ (0, T ; L p (Ω)) by Lemma 2.2. Hence Choosing M such that for any > 0, w M (hM )T ≤ /2 and 2C C/M ≤ /2 and using that β j (u j ) → β(u D ) in L 2 (0, T ; L 2 (Ω)), the result follows.
Then multiplying by h δ (v j (x, t)), where h δ is a smooth function such that h δ ≥ 0, 0 ≤ h δ ≤ 1 and h δ (r) → sign (r) as δ → 0 and integrating in Ω , we obtain We recall the conditions on f and take δ → 0 to get d dt We can then make j → ∞ and η → 0 to deduce Finally by Gronwall's inequality and passing to the limit as |y| tends to 0, we conclude Since we also have that u D is uniformly bounded in L ∞ (Ω) for t > 0, we obtain the uniform boundedness in space. The uniform boundedness in time comes from Lemma 3.1.
We can then conclude with the main result from this section. Proof. We deduce from Theorem 3.2 that there exists χ such that as D → ∞. Now consider, where Ψ is as defined in (3). We know from the subdifferential theory in [7] that Then, in particular, for any w ∈ L ∞ (0, T ; L ∞ (Ω)), and any η > 0, Using the lower semicontinuity of u → Ψ(u), the strong convergence of β(u D ) and the weak star convergence of u D in L ∞ (η, T ; L ∞ (Ω)), we have and thus χ = β(u) a.e. in (0, T ) × Ω.
In summary, we have, for any η > 0, Now consider φ ∈ L ∞ (0, T ) independent of space, then, passing to the limit, Since, by Lemma 2.2, we have that u and hence also β(u) and f (u) are independent of space and we conclude, for any φ ∈ L ∞ (0, T ) and therefore u is the solution of the equation in (2). Moreover, suppose that ∂φ ∂t ∈ L 2 (0, T ) and φ(T ) = 0. Then, again passing to the limit, Hence, β(u(0)) = 2 β(u 0 ) and the result follows.
4. Global attractor of the limit problem. In the previous section, we determined that, as D goes to infinity, the solution of problem (4) converges to the solution of the following ODE We now prove the existence of a global attractor in R, which we denote by A ∞ , associated to this limit problem. We recall that a semigroup S(t) belongs to the class K if, for each t > 0, the operator S(t) is compact and it is B-dissipative if it has a bounded global B-attractor, as defined in [14].
Theorem 4.1. The problem (9) defines a semigroup of class K, which is Bdissipative and so there exists a global B-attractor A ∞ associated with it. Moreover, the attractor A ∞ is equal to the union of all the bounded complete trajectories in R.
Proof. We define S(t) : R → R by S(t) 2 u 0 = u(t) with u the unique solution of problem (9). We will show that S(t) is of class K and also it is B-dissipative. To that end, we multiply the equation (β(u)) t + f (u) = 2 g by sign (β(u)) = sign (u) to obtain, d dt |β(u)| + f (u) sign (u) = 2 g sign (u).
We conclude that for each t > 0, S(t) maps bounded sets to bounded sets. As a result, S(t) is compact in R and thus has a maximal compact invariant global Battractor A ∞ , given as the union of all bounded complete trajectories in R.

5.
Continuity of attractors. The following lemma and theorem follow as in [16]. For completeness, we give the details of the proof of Theorem 5.2. First, we have that if the elements of the global attractors A D converge, as D goes to infinity, to a function, then such function must be equal to its spatial average. We can, at last, obtain the upper semicontinuity at infinity.  (4) and (9) respectively. For each t j > j, t 1 < t 2 < . . . < t j < . . ., consider x j ∈ A Dj such that S Dj (t j )(x j ) = v Dj 0 and then