REGULARITY OF GLOBAL ATTRACTORS FOR REACTION-DIFFUSION SYSTEMS WITH NO MORE THAN QUADRATIC GROWTH

. We consider reaction-diﬀusion systems in a three-dimensional bounded domain under standard dissipativity conditions and quadratic growth conditions. No smoothness or monotonicity conditions are assumed. We prove that every weak solution is regular and use this fact to show that the global attractor of the corresponding multi-valued semiﬂow is compact in the space ( H 10 (Ω)) N .

1. Introduction. In this paper we consider reaction-diffusion systems in a threedimensional bounded domain under dissipativity and growth conditions which guarantee global resolvability but not uniqueness of the Cauchy problem. In this general case it is known [8], [4] that all weak solutions generate a multi-valued semiflow, which has in the phase space H = (L 2 (Ω)) N an invariant, compact, connected, stable global attractor, which consists of bounded complete trajectories.
Some additional information becomes available if we consider the global attractor as a section of a trajectory attractor [1], [9], because a trajectory attractor is compact in the topologies of certain functional spaces [17], [3]. But more detailed information about the structure and regularity of the global attractor in the nonuniqueness case can be obtained only under additional assumptions on the system. In particular, the boundedness in H 1 0 (Ω) for the global attractor of the multivalued semiflow generated by regular solutions was obtained in [4] for scalar reactiondiffusion equations. Under an additional assumption on the external force it was shown in [4] that the global attractor of the multivalued semiflow generated by all weak solutions is bounded in (L ∞ (Ω)) N . In [10] the regularity of the global attractor was considered for a parabolic inclusion with linear growth of the multi-valued right-hand part. For the nonautonomous case see [6]. The compactness of the global attractor in H 1 0 (Ω) was obtained recently in [7] for a scalar reaction-diffusion equation with no more than quadratic growth. In this paper we extend the results from [7] for general reaction-diffusion systems with no more than quadratic growth.
The key point in order to prove this result is to obtain that any weak solution is regular. The question about the regularity of solutions for reaction-diffusion equations and inclusions without uniqueness is an important and interesting task, which has been studied recently in several works (see e.g. [2], [5], [7], [18]).

2.
Setting of the problem. In a bounded domain Ω ⊂ R n , 1 ≤ n ≤ 3, with sufficiently smooth boundary ∂Ω, we consider the following reaction-diffusion system (RD-system for short) and for given numbers C 1 , C 2 ≥ 0, γ > 0, p i ≥ 2, i = 1, N the following conditions hold: where 1 pi + 1 qi = 1, i = 1, N . Further, we shall use the notation H = (L 2 (Ω)) N , V = (H 1 0 (Ω)) N . It is well-known [1] that under conditions (2) for every initial data u 0 from the phase space H there exists at least one weak solution of (1) such that u(·) ∈ L p loc (0, +∞; L p (Ω)) ∩ L 2 loc (0, +∞; V ), p = (p 1 , ..., p N ), with u(0) = u 0 , and every weak solution of (1) belongs to C ([0, +∞); H), the function t → u(t) 2 H is absolutely continuous, for a.a. t ≥ 0 the following energy equality holds and for all t ≥ s ≥ 0 the following estimates hold where the positive constants C 3 , C 4 , δ depend only on the parameters of the problem (1). Moreover, the multivalued mapping G : is a multivalued semiflow (m-semiflow) [8], [9], which has the compact, invariant global attractor A ⊂ H, that is, a compact (in H) set A such that The main goal of this paper is to prove the compactness of A in the space V = (H 1 0 (Ω)) N and the attraction property (7) in the topology of V .
3. Main results. Our additional assumptions on the vector-function f are the following: the numbers p i from (2) are equal to 2 or 3 and there exists C > 0 such that for all v ∈ R N one has So we will consider the case of no more than quadratic growth of the non-linear term f . Let us consider some examples [14], [1], which show that assumptions (8) are not restrictive.
In this case p 1 = p 2 = 3, p 3 = 2 and conditions (8) are satisfied. In this example, the second condition in (2) is not satisfied. However, as shown in [14] the set for some a, b, c which depend on the constants of the problem, is an invariant region. Then restricting the solutions to this set we can prove that (2) holds.
In this case p 1 = p 2 = p 3 = 3 and conditions (8) are satisfied. It is clear that conditions (2) are also satisfied.
We denote by A the operator −∆ with Dirichlet boundary conditions, so that D (A) = H 2 (Ω) ∩ H 1 0 (Ω) . As usual, denote the eigenvalues and the eigenfunctions of A by λ i , e i , i = 1, 2...We define the usual sequence of spaces where α ≥ 0. It is well known that V s ⊂ H s (Ω) for all s ≥ 0 (see [16,Chapter IV] or [11]). We note also that V 1 = H 1 0 (Ω), V 0 = L 2 (Ω). The main result is the following.  Proof. The key idea is to prove that under condition (8) every weak solution u(·) is regular, that is Let us fix a weak solution u = (u 1 , ..., u N ), u(0) = u 0 and consider the first equation from (1) as an equation with unknown function v = u 1 and known fixed functions where g(t, x, v) = f 1 (v, u 2 (t, x), ..., u N (t, x)).
It should be noted that the problem (9) can have more than one solution and that a solution of (9) is not necessarily the first component of some solution of (1).
But u 1 is a solution of (9) and all facts which we will be able to prove for solutions of (9) will be true for u 1 . Therefore, in further arguments by a solution of (9) we always mean the function v = u 1 .
We will use the following well-known result.
The last inequality together with the weak convergence implies that As the global attractor A is invariant and bounded in H, then for sufficiently large R > 0 and, therefore, A is precompact in V . But A is closed in H, so it is closed in V and, finally, it is compact in V . Let us prove the attraction property (7) in the topology of V . If it is not true, then there exist δ > 0, R > 0 and sequences t n ∞, y n ∈ G(t n , B R ) such that From the dissipation property (5) But the global attractor consists of all cluster points (in H) of sequences ξ n ∈ G(t n , B R ) for all t n ∞ and R > 0. Thus up to subsequence y n → y ∈ A in H, and, therefore, in V , which is a contradiction with (25). The theorem is proved. Proof. Let a = ((a ij )), where a ij = 0, if i < j. Then the first equation in (1) is analyzed without any changes, the second one contains a summand a 21 ∆u 1 , which belongs to L 2 ( , T ; L 2 (Ω)) from (18), and so on. Lemma 3.6. In Examples 3.1-3.2 we can prove that A ⊂ (C ∞ (Ω)) N .