In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity satisfying the polynomial growth of arbitrary 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 -global attractors immediately; moreover, such an attractor can attract every bounded subset of with the -norm for any .
Let be a bounded domain with smooth boundary, we consider the long-time behavior for the solutions of the following semi-linear degenerate parabolic equation:
Let , then there exist constants such that
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 (see  for details) for equation (1.1) involving the Grushin operator. In , the authors extended the result for the -Laplacian. They considered as a self-adjoint operator and showed that generates an analytic semigroup in . They not only proved the existence of the global attractors in together with its fractal dimension but also showed the convergence of solutions to an equilibrium solution as the time . In the following, the authors of  have considered the case of critical growth nonlinearity and obtained the global attractors in by applying the decomposition technique introduced in .
In this paper, similar to the reaction–diffusion equations in [8,9], we consider equation (1.1) with the nonlinearity satisfying the polynomial growth of arbitrary order, i.e., . For our problem, we will confront two main difficulties when we establish the asymptotic higher-order integrability of solutions and the existence of the -global attractors. One difficulty is that the external force term with belongs only to , which leads to the fact that the solutions of equation (1.1) are only bounded in and do not have any higher regularity than the order (where is the homogeneous dimension of with a group of dilations corresponding to the -operator, see  for details). The other difficulty is that the Sobolev embedding theorem is not any longer valid since the growth exponent is arbitrary 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 -global attractors immediately (see Corollary 4.4). Moreover, the -global attractors indeed can attract every bounded subset of with the -norm for any . Finally, we also obtain the -global attractors (see Theorem 5.3).
2.1 The -operator
, , .
For every , the function , , where
There exists a constant such that
There exists a group of dilations which satisfy
This implies that the operator is -homogeneous of degree two, i.e.,
We will denote by the homogeneous dimension of with respect to the group of dilations , that is,
We have another expression of operator from the property (1) as (e.g., see )
2.2 Functional settings
For a function , we define
For any , , the embedding
In particular, for any , there exists a constant such that
For later application, we recall the following results (see  for details).
For any and any , the following equality holds:
Let be three Banach spaces satisfying with continuous embedding and suppose that there exist and a constant such that . Then, for any bounded sets , we have
Similar to Theorem 3.2 in , we have the following theorem.
Let be three Banach spaces satisfying with continuous embedding, be a semigroup defined on . Moreover, we assume that
has a global attractor in ;
is the solution of the stationary equation for (1.1);
there exist a family of operators defined on satisfying(2.4)
is a bounded absorbing set for the semigroup ;
if is closed in , then(2.6)
For any with and any bounded set in , from (2.4) we have
Given the fact that , (ii) is a direct result of the invariant of and (i):
In the following, we prove (iii).
Then, from the assumption we know that
2.3 Solutions for equations (1.1)
(Weak solutions) A function is said to be a weak solution of equation (1.1) if
(Strong solutions) A weak solution of equation (1.1) is said to be a strong solution if
3 -global attractors
3.1 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 ). Here we only state the results.
By Lemma 3.1, we can define the semigroup in as follows:
Moreover, we also have the following lemma about strong solutions, which can be obtained as that in  (similar to Theorem 8.5 in p. 227).
3.2 Some results
At first, similar to that in  (see Proposition 11.1 in p. 286 and Theorem 11.4 in p. 290), we have the following lemmas, which will be used to obtain the bounded absorbing sets and the global attractors in .
Next, similar to Theorem 3.4 in , we have the following lemma, which will be used to obtain the bounded absorbing sets in and .
4 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)
For this purpose, we decompose equation (1.1) as a stationary equation
Then, we consider the asymptotic higher-order integrability of equation (4.2).
Note that is dense in ; then for each and every initial data , we can find and such that
Consider the following approximation equations:
Set , then satisfies the following equation in distribution sense:
Moreover, we have
Hence, for any , we can take 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.
For each , there exist two positive constants and , which depend only on , , , and , such that for any , the solutions of (4.7) satisfy
At first, by taking the -inner product between equation (4.5) and , we know that there exists a positive constant which depends only on the -bounds of such that for every , the solution of (4.5) satisfies
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 , there are positive constants (that depends only on the bounds of ) and (that depends only on the bounds of ) such that
In the following, we will complete our proof.
(1) For the case .
(2) Assume that and hold for .
(3) We need to prove that and hold.
Let , we integrate (4.11) over and obtain that
Moreover, similar to Lemmas 3.2 and 3.3 in , we have the following results.
Moreover, for any , there is a subsequence of such that
4.2 -global attractors
In the following, we will establish the main results of this paper as follows.
Note that the homogeneous dimension , we have and then
Therefore, for any , we can take large enough such that
Now, let , be the strong solution of (4.4) and (4.5), respectively, then for any , by Lemma 4.2, we can find such that , for a.e. as . Then, by applying the Fatou lemma to in Theorem 4.1, we obtain that
As a directly application of Theorem 4.3, we have the following corollary.
Under the assumptions of Lemma 3.4, the semigroup defined by (3.1) has a -global attractor . Moreover, can attract every bounded subset of with -norm for any , and allows the decomposition ; here is bounded in for any , is a fixed solution of (4.1).
Moreover, by applying the interpolation inequality and (4.15), we can get that
On the other hand, from in Theorem 4.1, there are positive constants and such that for any , we have
5 -global attractors
In this section, we will prove the existence of the -global attractors.
At first, we need the following lemma.
Under the assumptions of Lemma 3.4, then for any bounded subset in , there exists a , depending only on the -norm of , such that
We give below only formal derivation of the estimate (5.1), which can be justified by the Galerkin approximation method.
By differentiating (1.1) with respect to time and denoting , we can get
On the other hand, multiplying (1.1) by , we obtain that
Set , then satisfies the following equation in distribution sense:
Taking the -inner product between (5.6) and , we have
The authors would like to thank the referee for his/her helpful comments and suggestions.
Funding information: This work was supported by the Natural Science Foundation of Hunan Province (Grants No. 2018JJ2416), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20C1263) and the Science and Technology Innovation Team of Hunan University of Arts and Science (Numerical calculation and stochastic process with their applications).
Conflict of interest: Authors state no conflict of interest.
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