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
where ( ) is a bounded domain with smooth boundary Γ, ν denotes the outer unit normal on Γ, , the nonlinearity f and the external force g satisfy some conditions specified later.
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:
under the assumptions that f, g satisfy the polynomial growth condition with order , and satisfies some weak assumption
for all , where θ is some positive constant.
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.
(H1) The functions and satisfy
for some , and
where ( ), , .
(H2) The external force is locally Lipschitz continuous, , and satisfies
(H3) Furthermore, is uniformly bounded in with respect to , i.e., there exists a positive constant K such that
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.
In order to study the problem (1)-(5), we recall the Sobolev space defined as the closure of in the norm
and denote by the dual space of X. We also define the Lebesgue spaces as follows:
for . Moreover, we have
for any , where the measure on is defined for any measurable set by . In general, any vector will be of the form with and , and there need not be any connection between and .
and let the operator be defined as follows:
Next, we recall briefly some lemmas used to prove the well-posedness of the solutions and the existence of the uniform (w.r.t. ) attractors for (1)-(3) under some assumptions on f.
Let be a bounded domain in and , let be given. Assume that , whereCis independent ofn, , as , almost everywhere in , and . Then , as weakly in .
Let and be the standard scalar product in . Then, for any , there exist two positive constants , , which depend onp, such that
Let and Ω be a bounded subset of with smooth boundary Γ. Then the inclusion
is compact for any , where
LetAbe defined in (7) and . Then, for any , one has
Furthermore, if and only if a.e. in .
LetXbe a given Banach space with dual , and letuandgbe two functions belonging to . Then the following three conditions are equivalent:
(i) uis almost everywhere equal to a primitive function ofg, i.e.,
for almost every ;
(ii) For each test function ,
(iii) For each ,
in the scalar distribution sense on .
If (i)-(iii) are satisfied, uis almost everywhere equal to a continuous function from intoX.
3 The well-posedness of solutions
In what follows, we assume that is given.
Definition 3.1 A function is called a weak solution of (1)-(3) on if
for all test functions .
Theorem 3.1Let Ω be a bounded domain in ( ). Assume thatfsatisfies (H1), is locally Lipschitz continuous and . Then, for any , any initial data and any , there exists a unique weak solution of (1)-(3), and the mapping
is continuous on .
Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin method (see ).
Consider the approximating solution in the form
where is an orthogonal basis of , which is included in . We get from solving the following problem:
Since f is continuous and g is locally Lipschitz continuous, using the Peano theorem, we get the local existence of . Next, we establish some a priori estimates for . We have
Thanks to (5), we obtain
by virtue of the following inequality (see Theorem 2.3.1 in ):
Let and , we deduce from (10) and (12) that
Integrating (13) over , we obtain
for any .
Due to (14), we get
Therefore, is uniformly bounded in n in the , , respectively, and is uniformly bounded in n in the , and one can extract a subsequence of such that
Let be a projection. For any , set , we have
We perform the following estimate deduced from the Hölder inequality and the Young inequality:
Using the boundedness of in again, we infer that
Since , , , we find
Therefore we can extract a subsequence such that
By virtue of the Aubin compactness theorem, we can extract a further subsequence (still denoted by ) such that additionally
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
for any .
In order to prove that u is a weak solution of (1)-(3), it remains to show that . Noticing that
it follows from the formulation of and that in and in . Moreover, by the lower semi-continuity of and , we obtain
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
Combining (24) with in , we obtain
Therefore, from Lemma 2.2, the Hölder inequality and the Young inequality, we deduce that for any ,
which implies that in , hence .
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
Therefore, a.e. in if in , and is continuously dependent on the initial data.
by use of Lemma 2.5, we know that
Therefore, is meaningful. □
By Theorem 3.1, we can define a family of continuous processes in as follows: For all ,
where is the solution of (1)-(5) with initial data . That is, a family of mappings satisfies
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 set is called to be -uniformly (w.r.t. ) absorbing for if for any and any bounded subset , there exists a positive constant such that
for any .
A set is said to be -uniformly (w.r.t. ) attracting for the family of processes , if
for an arbitrary fixed and any bounded set .
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 : .
Define the uniform (w.r.t. ) ω-limit set of B by . This can be characterized by the following: if and only if there are sequences , , , such that ( ).
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.
If a family of processes is -uniformly asymptotically compact, then for any , ,
(i) for any sequences , , , as , there is a convergent subsequence of inY,
(ii) is nonempty and compact inY,
(v) ifAis a closed set and -uniformly (w.r.t. ) attractingB, then .
Assumption 1 Let be a family of operators acting on Σ and satisfying:
(i) , ,
(ii) translation identity:
The kernel of the process acting on X consists of all bounded complete trajectories of the process :
The set is said to be kernel section at time , .
A family of processes is said to be -weakly continuous if for any fixed , , the mapping is weakly continuous from to Y.
Assumption 2 Let Σ be a weakly compact set and be -weakly continuous.
Under Assumptions 1 and 2 with , which is a weakly continuous semigroup, if acting 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.
where is 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 at of kernel of the process with symbol . Furthermore, is nonempty for all .
Let be a family of processes on ( ) and suppose has a bounded -uniformly (w.r.t. ) absorbing set in . Then, for any , and any bounded subset , there exist two positive constants and such that
for any , , .
Let a family of processes be -uniformly (w.r.t. ) asymptotically compact, then is -uniformly asymptotically compact for , if
(i) has a bounded -uniformly (w.r.t. ) absorbing set ,
(ii) for any , and any bounded subset , there exist two positive constants and such that
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 .
If ℰ is reflective separable and , then
(i) for all , ,
(ii) the translation group is weakly continuous on ,
(iii) for ,
(iv) is weakly compact.
Due to Lemma 4.5, is weakly compact and the translation semigroup satisfies that and is weakly continuous on . Because of the uniqueness of solution, the following translation identity holds:
Theorem 4.1The family of processes corresponding to problem (1)-(5) is -weakly continuous and -weakly continuous.
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 :
for any .
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 . □
(The uniform Gronwall lemma) Let , , be three positive locally integrable functions on , and for some and all , , , satisfy the following inequalities:
whereR, A, Bare three positive constants. Then
for all .
4.2 The existence of uniformly absorbing sets
In this subsection, we prove the existence of uniformly (w.r.t. ) absorbing sets for the process corresponding to (1)-(5).
Theorem 4.2Assume thatfandgsatisfy (H1)-(H2). Then the family of processes corresponding to problem (1)-(5) has a bounded - and -uniformly (w.r.t. ) absorbing set. That is, for any bounded subsetBof and any , there exist , and two positive constants , such that
for any and
for any , where , , , and are specified in (33), (41), (32) and (40), respectively.
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
Let and , we deduce from (12) and (29) that
It follows from the classical Gronwall inequality and Lemma 4.5 that
where we have used the following inequality:
From (31), we deduce that
Integrating (30) over , we obtain
Let , we deduce from (5) that there exist three positive constants , , β such that
Thanks to (34), we deduce from (35)-(36) that