This paper studies the long-time asymptotic behavior of solutions for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions in n-dimensional bounded smooth domains. We have proved the existence of the uniform attractor in for the non-autonomous p-Laplacian evolution equations subject to dynamic nonlinear boundary conditions by using the Sobolev compactness embedding theory, and the existence of the uniform attractor in by asymptotic a priori estimate.
We are concerned with the existence of uniform attractors for the process associated with the solutions of the following non-autonomous p-Laplacian equation:
Equation (1) is subject to the dynamic flux boundary condition
and the initial condition
Non-autonomous equations appear in many applications in the natural sciences, so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (e.g., see [1-4]). The first attempt was to extend the notion of a global attractor to the non-autonomous case, leading to the concept of the so-called uniform attractor (see ). It is remarkable that the conditions ensuring the existence of a uniform attractor are parallel with those for the autonomous case. A uniform attractor need not be ‘invariant’, unlike a global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many non-autonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems.
Dynamic boundary conditions are very natural in many mathematical models such as heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two mediums, problems in fluid dynamics (see [1-4,6-11]).
In recent years, many authors have studied p-Laplacian equations (see [12-17]) and the problem (1)-(3) for (see [3,7,9,10]) by discussing the existence and uniqueness of local solutions, the blow-up of solutions, the global existence of solutions, the global attractors of solutions and the eigenvalue problems, etc. In , the authors have proved the global existence of solutions for quasi-linear elliptic equations with dynamic boundary conditions. Due to the complications inherent to nonlinear dynamic boundary conditions, these problems (1)-(3) still need to be investigated. In [15-17,19], the authors have considered the eigenvalue problem
and obtained some results, and some p-Laplacian elliptic equations with nonlinear boundary condition have been studied by using these results mentioned in [15-17,19]. In [14,20], the authors have proved the existence of uniform attractors for the non-autonomous p-Laplacian equations with Dirichlet boundary conditions in a bounded and an unbounded domain in . The authors have proved the existence of global attractors for the autonomous p-Laplacian equations with dynamic flux boundary conditions in . In , the authors have used a new type of uniformly Gronwall inequality and proved the existence of a pullback attractor in of the following equation:
Moreover, the existence of uniform attractors for the non-autonomous p-Laplacian equations with dynamical boundary conditions remains unsolvable.
To study problem (1)-(3), we assume the following conditions.
The main purpose of this paper is to study the long-time dynamical behavior for the non-autonomous p-Laplacian evolutionary equations (1)-(3) under quite general assumptions (4)-(6). We first prove the existence and the uniqueness of solutions for (1)-(5), and then the existence of uniformly (w.r.t. ) absorbing sets for the process corresponding to (1)-(5) in and , respectively, is obtained. Finally, the existence of the uniform (w.r.t. ) attractor for the process corresponding to (1)-(5) in is obtained by the Sobolev compactness embedding theory and the existence of the uniform (w.r.t. ) attractor for the process corresponding to (1)-(5) in is obtained by asymptotic a priori estimate.
This paper is organized as follows. In Section 2, we give some notations and lemmas used in the sequel. The existence and the uniqueness of solutions for the problem (1)-(5) have been proved in Section 3. Section 4 is devoted to proving the existence of the uniformly (w.r.t. ) absorbing sets in , and , respectively, for the process corresponding to (1)-(5) and the existence of the uniform (w.r.t. ) attractors in , and , respectively, for the process corresponding to (1)-(5).
Throughout this paper, we denote the inner product in (or ) by , and let C be a positive constant, which may be different from line to line (and even in the same line); we denote the trace operator by γ.
(i) uis almost everywhere equal to a primitive function ofg, i.e.,
3 The well-posedness of solutions
Theorem 3.1Let Ω be a bounded domain in (). Assume thatfsatisfies (H1), is locally Lipschitz continuous and. Then, for any, any initial dataand any, there exists a unique weak solutionof (1)-(3), and the mapping
Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin method (see ).
Thanks to (5), we obtain
by virtue of the following inequality (see Theorem 2.3.1 in ):
Due to (14), we get
We perform the following estimate deduced from the Hölder inequality and the Young inequality:
Therefore we can extract a subsequence such that
Due to the boundedness of in and (5), we obtain that is uniformly bounded in and hence in , similarly, in . By virtue of (16)-(17), we see that a.e. in and a.e. in , then a.e. in and a.e. in . Thanks to Lemma 2.1, we know that
Therefore, we have
Meanwhile, by the Lebesgue dominated theorem, one can check that
This fact and (20)-(21) imply
In view of (18), we have
This and (22) deduce
To this end, we first observe that
On the other hand, it follows from Lemma 2.4 that
Finally, we prove the uniqueness and continuous dependence of the initial data of the solutions. Let , be two solutions of (1)-(5) with the initial data , , respectively. Let . Taking the inner product of the equation with w, we deduce that
By virtue of (4) and Lemma 2.2, we obtain
which implies that
by use of Lemma 2.5, we know that
4 Existence of uniform attractors
In this section, we prove the existence of uniform attractors for (1)-(3).
4.1 Abstract results
In this subsection, let Σ be a parameter set, let X, Y be two Banach spaces, continuously. is a family of processes in a Banach space X. Denote by the set of all bounded subsets of X and . In the following, we give some basic definitions and some abstract results about the existence of bi-space uniform (with respect to (w.r.t.) ) attractors.
A closed set is said to be an -uniform (w.r.t. ) attractor for the family of processes if it is -uniformly (w.r.t. ) attracting and it is contained in any closed -uniformly (w.r.t. ) attracting set for the family of processes : .
A family of processes possessing a compact -uniformly (w.r.t. ) absorbing set is called -uniformly compact. A family of processes is called -uniformly asymptotically compact if it possesses a compact -uniformly (w.r.t. ) attracting set, i.e., for any bounded subset and any sequences , as and , is precompact in Y.
(ii) translation identity:
Under Assumptions 1 and 2 with, which is a weakly continuous semigroup, ifacting onXis-uniformly (w.r.t. ) asymptotically compact, then it possesses an-uniform (w.r.t. ) attractor, which is compact inYand attracts all the bounded subsets ofXin the topology ofY.
whereis a bounded neighborhood of the compact-uniformly attracting set inY; i.e., is a bounded-uniformly (w.r.t. ) absorbing set of. is the section atof kernelof the processwith symbol. Furthermore, is nonempty for all.
From Theorem 3.1, we know that the problem (1)-(5) generates a process acting in and the time symbol is . We denote by the space endowed with a locally weak convergence topology. Let be the hull of g in , i.e., the closure of the set in and .
Proof For any fixed and τ, , , let () weakly in and weakly in as , denote by . The same estimates for given in the Galerkin approximations (in Section 3) are valid for the here. Therefore, for some subsequence and such that for any , , weakly in and . And the sequence , is bounded in . Denote by , and the weak limits of , and in , and , respectively. So, we get the following equation for :
By the same method as the proof of Theorem 3.1, we know that , and , which means that in V is the weak solution of (1)-(5) with the initial condition . Due to the uniqueness of the solution, we state that weakly in and . For any other subsequence, and satisfy weakly in and , by the same process, we obtain the analogous relation weakly in and holds. Then it can be easily seen that for any weakly convergent initial sequence and weakly convergent sequence , we have weakly in and . □
whereR, A, Bare three positive constants. Then
4.2 The existence of uniformly absorbing sets
Theorem 4.2Assume thatfandgsatisfy (H1)-(H2). Then the family of processescorresponding to problem (1)-(5) has a bounded- and-uniformly (w.r.t. ) absorbing set. That is, for any bounded subsetBofand any, there exist, and two positive constants, such that
Proof Taking the inner product of (1) with u, we deduce that
By virtue of (5), the Hölder inequality and the Young inequality, we obtain
It follows from the classical Gronwall inequality and Lemma 4.5 that
where we have used the following inequality:
From (31), we deduce that
Thanks to (34), we deduce from (35)-(36) that
Combining (37) with (38), by virtue of the uniform Gronwall Lemma 4.6, we get
Theorem 4.3The family of processescorresponding to problem (1)-(5) with initial datahas an-uniform (w.r.t. ) attractor, which is compact inand attracts every bounded subsetBofin the topology of. Moreover,
for some positive constant c. Therefore,
From (42)-(45), we deduce that
Multiplying (46) by ζ and integrating over Ω, and combining (4) with (47), we obtain
Therefore, we deduce from the uniformly Gronwall inequality that
Theorem 4.5Assume thatfandgsatisfy (H1)-(H3). Then the family of processescorresponding to problem (1)-(5) with initial datais-uniformly (w.r.t. ) asymptotically compact, i.e., there exists a compact uniformly attracting set in, which attracts any bounded subsetin the topology of.
which combining with Corollary 4.1, Theorem 4.3 and Theorem 4.4 yields Theorem 4.5 immediately. □
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
The authors would like to thank the referees for their valuable suggestions.
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