LONG-TIME BEHAVIOR FOR A CLASS OF WEIGHTED EQUATIONS WITH DEGENERACY

. In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space H 1 ,a 0 (Ω) (see Deﬁnition 1.1) cannot be bounded embedded into L 2 (Ω). We claim that the dimension of the global attractor is inﬁnite by estimating its lower bound of Z 2 -index. Moreover, we prove that there is an inﬁnite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.


Introduction.
Let Ω ba a bounded smooth domain in R n (n ≥ 2) with smooth boundary ∂Ω, we consider the long time behavior of a class of degenerate parabolic equations − div(a(x)∇u) − λu + |u| p−2 u = 0, in Ω × R + , u = 0 on ∂Ω × R + , u(x, 0) = u 0 ∈ L 2 (Ω) in Ω, (1) where p ≥ 2 and a(x) satisfied the following assumption (A 1 ): a(x) ∈ L ∞ (Ω) and a(x) = 0 for x ∈ Σ, a(x) > 0 for x ∈ Ω\Σ, where Σ is a closed subset of Ω with meas(Σ) = 0. For our problem, it is natural to look for solutions in weighted Sobolev spaces H 1,a 0 (Ω) which is defined as follows. Definition 1.1. Let Ω be a bounded domain in R n and a(x) satisfy (A 1 ). H 1,a 0 (Ω) is defined as the completion of C ∞ 0 (Ω) with respect to the norm Furthermore, H 1,a 0 (Ω) is a Hilbert space with respect to the scalar product In addition, since our problem is mainly dependent on the properties of the norm of weighted Sobolev space H 1,a 0 (Ω), the following assumption is always needed. (A 2 ) The embedding H 1,a 0 (Ω) → L r (Ω) is compact for some r ∈ [1 + ∞). Usually, this assumption (A 2 ) can be easily satisfied. For example, (A 2 ) holds if a(x) satisfies which will be proved in Appendix A. Under the condition of (A 2 ), if r ≥ 2, the problem is completely similar to the usual reaction-diffusion equations which is a(x) ≡ 1. However, if 1 ≤ r < 2 and H 1,a 0 (Ω) can not even be bounded embedded into L 2 (Ω) the problem is very different from the usual case.
Remark 1. A model for the weight a is |x| β for 2 < β < N as α = n in (3). And it is obvious that there are a lot of functions u which belong to H 1,a 0 (Ω) but not to L 2 (Ω), for example, u(x) = α(x) |x| n/2 , x ∈ Ω and α(x) ∈ C 1 0 (Ω).
For degenerate parabolic PDEs when H 1,a 0 (Ω) cannot be bounded embedded into L 2 (Ω), not so much is known about the long-time behavior although there are a lot of work for it when H 1,a 0 (Ω) can be compactly embedded into L 2 (Ω) (see [1]- [14]). Here, we are particularly interested in the case that 1 ≤ r < 2 in (A 2 ) and H 1,a 0 (Ω) can not even be bounded embedded into L 2 (Ω) to consider the existence and properties of the global attractor for Eq. (1). The problem is far from being just technical and the degeneration in this case can lead to essentially new types of attractors which are not observable in 'regular' equation in bounded domains. As shown, the global attractors of the degenerate equations are infinite-dimensional.
Since Efendiev & Zelik have given the first example of a physically relevant dissipative system with an infinite-dimensional global attractor in [6], more and more attention is being paid to some dissipative system with an infinite-dimensional global attractor, see [6]- [9]. In order to study such attractors one usually uses the concept of Kolmogorov's ε-entropy, see [6]- [10]. For some symmetric dynamical system with Lyapunov function in a Banach space, some authors in [15,19,21,22,20] have shown that Z 2 -index is a useful tool to estimate the lower bounded of the dimension of the attractor. In this paper, we consider the degenerate case of (1) using the symmetry of its Lyapunov function, as shown, which provides another example that the dimension of global attractor is infinity. First, we prove the existence of the global solution and the global attractor of (1). And then we use Z 2 index theory to show that the dimension of the global attractor of (1) is infinite. Finally, we prove that there is an infinite sequence of equilibrium points in the global attractor, in particularly which converges to 0. This paper is organized as follows. In section 2, we prove the existence and uniqueness of solution, and in section 3 we further prove the existence of a global attractor. In section 4 we estimate the lower bound of Z 2 -index of a subset of the global attractor and show that its dimension is infinite. Finally we show in section 5 that there exists an infinite sequence of stationary points which goes to 0 and that the critical value sequence of the Lyapunov functional also goes to 0.
2. Existence of the global solutions. We now study the existence of the weak solutions of Eq. (1) defined by the integral equality in a weighted Sobolev space. For convenience, throughout we denote Ω T = Ω×[0, T ], V = L 2 (0, T ; H 1,a 0 (Ω))∩L p (Ω T ) and V * the dual space of V , respectively. In addition, let · p be the norm of L p (Ω)(p ≥ 1), |u| be the modular(or absolute value) of u, C be the arbitrary positive constants, which may be different from line to line and even in the same line.
According to the classical theory on parabolic equations (see for example [4,5,17,18]), we know the problem admits a unique weak solution and u ε | t=0 = u ε,0 almost everywhere in Ω. We do some estimates on u ε in the following. Multiplying (4) by u ε and integrating over Ω, we get 1 2 We can use Young's inequality to write 1 2 where |Ω| = Ω dx, The Gronwall lemma implies u ε is uniformly bounded in L ∞ (0, T ; L 2 (Ω)) with respect to ε for any T ∈ R. So a subsequence still denoted as u ε and u in L ∞ (0, T ; L 2 (Ω)) can be found such that Integrating (7) both sides between 0 and T , we may get We now extract a weakly convergent subsequence, denoted also by u ε for convenience, with Therefore, to obtain the existence, it suffices for us to show From (8) we can infer that as ε → 0. Multiplying (4) by ϕ and letting ε → 0 + it is immediate that (9) holds and u is a weak solution of Problem (1).
To prove uniqueness and continuous dependence, let u 0 and v 0 be in H 1,a 0 (Ω) ∩ L p (Ω) and consider w(t) = u(t) − v(t). Then and multiplying by w and integrating over Ω gives 1 2 Note that there exists a constant C > 0 such that shows that 1 2 Integrating this gives the uniqueness and continuity in L 2 (Ω). Now we can use these solutions to define a semigroup {S(t)} t≥0 in L 2 (Ω), which is a closed bounded absorbing set in L 2 (Ω), by setting which is continuous on u 0 in L 2 (Ω).
3. Existence of the global attractor. Our goal in this section is to show that the semigroup associated with the degenerate equation (1) possesses the global attractor in L 2 (Ω). We start with the existence of bounded absorbing sets.
where both ρ 0 and ρ 1 are positive constants independent of B, u(t) = S(t)u 0 .
Proof. Multiplying (1) by u, integrating over Ω, we get d dt Note that p ≥ 2, from (11) we have d dt Using the Gronwall lemma, there exists a constant T 0 ( u 0 H 1,a 0 ∩L p ), such that u(t) 2 2 ≤ ρ 0 for t ≥ T 0 . Taking t ≥ T 0 and integrating (11) on [t, t + 1] gives With an application of the existence of the bounded absorbing sets in L 2 (Ω), it follows that for any t ≥ T 0 , On the other hand, multiplying (1) by u t and integrating over Ω, we obtain By the uniform Gronwall lemma, (13) implies that for t ≥ T 0 + 1 Then we have Ω a(x)|∇u| 2 dx + 2 p Ω |u| p dx ≤ C(λ, ρ 0 ). Now, taking ρ 1 = C(λ, ρ 0 ) and T = T 0 + 1, we complete the proof of Theorem 3.1.
By the uniform compactness method introduced in [4,5,17,18] and H 1 0 (Ω) is compactly embedding in L r (Ω), we know that the absorbing set B in H 1 0 (Ω) is compact in L r (Ω). Thanks to the L p (Ω) interpolation inequality u L 2 ≤ u p(r−2)/2(r−p) Then inf u∈M ∩X ⊥ n Ω a(x)|∇u| 2 dx = 0 holds for any finite dimensional subspace X n of X, where X ⊥ n = {u ∈ X | Ω uvdx = 0, ∀v ∈ X n } and X is endowed with the norm u X = u 2 Proof. Assume by contradiction there exist some n 0 and an n 0 dimensional subspace X n0 of X such that inf u∈M ∩X ⊥ n 0 Ω a(x)|∇u| 2 dx = α > 0, which implies that X ⊥ n0 with the norm of H 1,a 0 (Ω) is bounded embedded into L 2 (Ω), that is, So X ⊥ n0 is a closed subspace of L 2 (Ω) ∩ H 1,a 0 (Ω) with respect to the norm of H 1,a 0 (Ω) and X ⊥ n0 is a finite codimensional subspace of X. Let This contradicts the assumption (14).
Lemma 4.2. Let X and M be defined as in Lemma 4.1. Suppose that H 1,a 0 (Ω) cannot be bounded embedded into L 2 (Ω), then for any ε > 0 and any n there exists some n-dimensional subspace X n of X, such that sup u∈Xn∩M Ω a(x)|∇u| 2 dx < ε.
Proof. For any fixed n and any ε > 0, in view of the assumption that H 1,a 0 (Ω) cannot be bounded embedded into L 2 (Ω), u 1 ∈ M can be chosen such that Continuing inductively in this way, let X i−1 =span{u 1 , . . . , u i−1 } (1 < i ≤ n+1), and u i can be chosen in M ∩ X ⊥ i−1 such that Ω a(x)|∇u i | 2 dx < ε n .
Then for any u ∈ X n ∩ M , which can be written to be u = n i=1 α i u i and Hence, for each i, The proof is complete. Now we estimate the attractor's lower bound of Z 2 -index, and then show that the dimension of the attractor of (1) is infinite. Theorem 4.3. Suppose that H 1,a 0 (Ω) cannot be bounded embedded into L 2 (Ω) and A is the global attractor of (1), then for any n ∈ Z there exists an α n > 0, which is small enough, such that Proof. For any fixed n ∈ Z, from Lemma 4.2 for ε = λ 2 there exists the n-dimensional subspace X n of X such that where M is from Lemma 4.1, and which implies that for any u ∈ X n . So for any u ∈ X n we have Since each equipped norm in a finite dimensional subspace is equivalent, the norms ( Ω u 2 dx) 1 2 and ( Ω u p dx) 1 p are equivalent refined in X n , that is, there exist C 1 , C 2 > 0 such that Hence, Thus there exist an α n > 0 and a δ > 0 small enough such that E(u) ≤ −α n when u ∈ X n and u L 2 (Ω) = δ.
In virtue of the decreasing of E(S(t)u) in t, we know It follows that The monotonicity of Z 2 -index follows from (16) and (17) that Hence the proof is complete.
Note that any compact set E with fractal dimension dim F E = n can be mapped in to R 2n+1 by a linear (odd) Hölder continuous one-to-one projector, see [11]. So we have the following result. Corollary 1. Under the above assumptions, the fractal dimension of the global attractor dim A = +∞.

5.
Multiple equilibrium points in the global attractor. In this section, we show the existence of multiple equilibria in the global attractor for our problem (1). We will need the following deformation lemma in [22]. Lemma 5.1 ([22]). Let {S(t)} t≥0 be a continuous semigroup on Banach space X. Assume that S(·) possesses a C 0 even Lyapunov function E on X and a global attractor A. Set K = {u ∈ A : S(t)u = u, ∀t ≥ 0} and K c = {u ∈ K : E(u) = c} . Then for any δ neighborhood N (K c , δ) of K c , there exist ε > 0 and t ε > 0 such that where Γ k = {A ⊂ A | A is closed , A = −A and γ(A) ≥ k}. Then (i) For each c k , there exists some equilibrium points u * k ∈ X of (1) such that To prove (i), we suppose by contradiction that there exists a k 0 such that for any u ∈ A and S(t)u = u but E(u) = c k0 , that is, K c k 0 = ∅. Then Lemma 5.1 implies that there exist some ε > 0 and t ε > 0 such that In addition, by the definition of c k0 , there exists some A ∈ Γ k0 such that On the other hand, it follows from γ(S(t ε )A) ≥ γ(A) ≥ k 0 that S(t ε )A ∈ Γ k0 . Also by the definition of c k0 , we know that which contradicts (19).
(ii) is immediate since γ({0}) = ∞ and Γ i+1 ⊂ Γ i . To prove the third assertion, we also suppose by contradiction that γ(K c k 0 ) ≤ p. Then there exist a δ neighborhood N (K c k 0 , δ) of K c k 0 such that since K c k 0 is compact. By the definition of c k0+p , there exists an A ⊂ Γ c k 0 +p such that On the one hand, the sub-additivity of Z 2 -index implies that On the other hand, Lemma 5.1 implies that there exist some ε > 0 and t ε > 0 such that So (21) and (22) imply which contradict the definition of c k0 . So γ(K c k 0 ) ≥ p + 1 ≥ 2 (23) as p ≥ 1. It follows that the semigroup {S(t)} t≥0 possesses infinite pairs of different fixed points in K c k 0 , then there are infinite equilibria in A ∩ E −1 ((−∞, 0)).
Next we will show that β k → +∞ as k → ∞. Set N k = k i=1 ker f i , then M = N k ⊕ span{ϕ 0 , · · · , ϕ k−1 }. From the property of Z 2 -index of a set, we know that A ∩ N k = ∅ if A ⊂ M 0 and γ(A) ≥ k + 1. Theorem 5.4. c k → 0 as k → ∞ where c k is given by (18). Furthermore, u * k → 0 in L p (Ω) as k → ∞.
Proof. From the proof of Theorem 5.2, for each c j , there exists some equilibrium points u * j ∈ X of (1) such that