Global attractors for a class of semilinear degenerate parabolic equations

Abstract: In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f satisfying the polynomial growth of arbitrary − p 1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( ( ) ( )) L L Ω , Ω p 2 -global attractors immediately; moreover, such an attractor can attract every bounded subset of ( ) L Ω 2 with the + L p δ-norm for any ∈ [ +∞) δ 0, .

Abstract: In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f satisfying the polynomial growth of arbitrary − p 1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( ( ) ( )) L L Ω , Ω where Δ λ is the degenerate elliptic operator, which will be characterized in Section 2. The external forcing ( ) = ∑ + ( ) ∈ ( ) , the nonlinearity ∈ ( ) f C , 1 and satisfies the following classical assumptions (e.g., see [1][2][3] The long-time behavior for the solutions of semi-linear degenerate parabolic equations has been considered by many researchers, e.g., see [4][5][6] and references therein. For the subcritical growth case, the authors have established the existence of the global attractors in ( ) S Ω 0 1 (see [4] for details) for equation (1.1) involving the Grushin operator. In [5], the authors extended the result for the Δ λ -Laplacian. They considered Δ λ as a self-adjoint operator and showed that −Δ λ generates an analytic semigroup in ( ) L Ω 2 .
They not only proved the existence of the global attractors in ( ) W Ω λ 1,2 together with its fractal dimension but also showed the convergence of solutions to an equilibrium solution as the time → ∞ t . In the following, the authors of [6] have considered the case of critical growth nonlinearity and obtained the global attractors by applying the decomposition technique introduced in [7]. In this paper, similar to the reaction-diffusion equations in [8,9], we consider equation (1.1) with the nonlinearity f satisfying the polynomial growth of arbitrary − p 1 order, i.e., ( ) −| | − f u u ũ p 2 ( ≥ ) p 2 . For our problem, we will confront two main difficulties when we establish the asymptotic higher-order integrability of solutions and the existence of the ( ( ) ( )) L L Ω , Ω p 2 -global attractors. One difficulty is that the external force (where Q is the homogeneous dimension of N with a group of dilations corresponding to the Δ λ -operator, see [5] for details). The other difficulty is that the Sobolev embedding theorem is not any longer valid since the growth exponent is arbitrary − ( ≥ ) p p 1 2 order. Based on the aforementioned difficulties and motivated by the idea of [10][11][12][13], we first decompose equation (1.1) as a stationary equation and an evolutionary equation, then establish some asymptotic higher-order integrability results about the difference of the solutions near the initial time by using the bootstrap method (see Theorems 4.1 and 4.3). As an application, we obtain the ( ( ) ( )) L L Ω , Ω
(4) There exists a group of dilations ( ) > δ r r 0 which satisfy This implies that the operator Δ λ is δ r -homogeneous of degree two, i.e., We will denote by Q the homogeneous dimension of N with respect to the group of dilations ( ) > δ r r 0 , that is, We have another expression of operator Δ λ from the property (1) as (e.g., see [5]) Similar to that in [5,17], we have the following embedding theorem.
In particular, for any ∈ ( ) and the optimal constant of C is denotes the first eigenvalue of the operator −Δ λ on Ω with homogeneous Dirichlet boundary conditions.
For later application, we recall the following results (see [12] for details).
and any > r 0, the following equality holds: where ⋅ stands for the usual inner product in N .
Similar to Theorem 3.2 in [12], we have the following theorem.
S be a semigroup defined on X. Moreover, we assume that (a) (⋅) S has a global attractor A in X; for any t 0, 1 and a bounded set B 0 in Z satisfying that for any ≥ t 0 and any bounded set B in X there exists a = ( ) Then the following conclusions hold: Proof. For any ≥ t T , 0 with ≥ t T and any bounded set B in X, from (2.4) we have Given the fact that In the following, we prove (iii). At first, (2.6) follows directly from (2.5) and the assumption that B 0 is closed in X. Then, from the assumption ( ) c we know that and consequently, by applying Lemma 2.3, we obtain that

Solutions for equations (1.1)
In this subsection, being similar to the reaction-diffusion equations, we will give the definition of different solutions about equation (1.1) (see [2] for details).

.1 Existence and uniqueness of solutions
In this subsection, we first give the existence and uniqueness of weak solutions, which can be obtained by the Fadeo-Galerkin method (similar to Theorem 8.4, p. 221 in [2]). Here we only state the results.
Moreover, we also have the following lemma about strong solutions, which can be obtained as that in [2] (similar to Theorem 8.5 in p. 227).
Then, for any > T 0, there exists a unique strong solution u to equation (1.1) such that Similar to [12,13], we need the following lemma about the strong solutions, which guarantees that the test functions that we used in the following are meaningful.

Some results
In this subsection, we give some results about the solutions of equation (

Asymptotic higher-order integrability
In this section, we will establish some asymptotic higher-order integrability for the solutions of equation (1.1).
4.1 Some a priori estimate for the solutions of (4.7) In this subsection, based on the Alikakos-Moser iteration technique (see [18]), we will establish the following induction estimates about the solutions of (4.7).
For this purpose, we decompose equation (1.1) as a stationary equation and an evolutionary equation By [19] we know that equation and by [19] we know that equation (4.5) has a strong solution v n satisfying t n λ n n n n n n n 0 (4.7) Moreover, we have and so for any ≥ r 0, ,˚Ω Ω f o r a . e . 0 , .
Hence, for any ≥ r 0, we can take | (⋅ )| ⋅ (⋅ ) w t w t , , n r n as a test function for equation (4.7) (this is the main reason for the approximations above).
With the preparation above, we have the following main result of this subsection. , such that for any = … n 1, 2, , the solutions w n of (4.7) satisfy where Q is the homogeneous dimension of N with a group of dilations corresponding to the Δ λ -operator.
Proof. At first, by taking the L 2 -inner product between equation Second, for the solution of (4.4), similar to the estimate for the solution of equation (1.1) (e.g., Lemma 3.6), we know that for each ∈ n , there are positive constants ∥ ∥ S g n i 2 (that depends only on the bounds of ∥ ∥ g n i 2 ) and ∥ ∥ T u n 0 2 (that depends only on the bounds of ∥ ∥ u n 0 2 ) such that In the following, we will complete our proof.
(2) Assume that ( ′) A k and ( ′) B k hold for ≥ k 0. and integrating over ∈ x Ω, then we obtain that

-global attractors
In the following, we will establish the main results of this paper as follows.  Here, the constant Λ p Q δ , , depends only on p Q δ , , , and ( ) v x is one of the fixed (independent of δ) solutions of (4.1).
Proof. Note that the homogeneous dimension ≥ Q 3, we have > − 1 Q Q 2 and then Therefore, for any ∈ [ ∞) δ 0, , we can take k large enough such that Now, let u n , v n be the strong solution of (4.4) and (4.5), respectively, then for any ≥ t T k , by Lemma 4.2, we can find n j such that for a.e. ∈ x Ω as → ∞ j . Then, by applying the Fatou lemma to ( ′) A k in Theorem 4.1, we obtain that Therefore, combining with (4.15) and the ∩ W L λ p 1,2 -dissipation (see Lemma 3.6), applying the interpolation inequality, we can define B δ as follows: and (4.14) follows from (4.15) immediately. □ As a directly application of Theorem 4.3, we have the following corollary.
for all ≥ t T k1 , where we have used (2.5) in Theorem 2.4 and the attraction of ( ( ) ( )) L L Ω , Ω 2 2 -global attractors, B is a L 2 -bounded set.