Uniform attractors for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions
© Li and You; licensee Springer 2013
Received: 27 December 2012
Accepted: 1 May 2013
Published: 17 May 2013
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.
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]).
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.
where (), , .
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.
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 .
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.
Lemma 2.1 
Let be a bounded domain in and , let be given. Assume that , where C is independent of n, , as , almost everywhere in , and . Then , as weakly in .
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
Furthermore, if and only if a.e. in .
Lemma 2.5 
- (i)u is almost everywhere equal to a primitive function of g, i.e.,
- (ii)For each test function ,
- (iii)For each ,
in the scalar distribution sense on .
If (i)-(iii) are satisfied, u is almost everywhere equal to a continuous function from into X.
3 The well-posedness of solutions
In what follows, we assume that is given.
for all test functions .
is continuous on .
Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin method (see ).
for any .
for any .
which implies that in , hence .
Therefore, a.e. in if in , and is continuously dependent on the initial data.
Therefore, is meaningful. □
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.
for any .
for an arbitrary fixed and any bounded set .
Definition 4.2 
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 : .
Definition 4.3 
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 ().
Definition 4.4 
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.
Lemma 4.1 
for any sequences , , , as , there is a convergent subsequence of in Y,
is nonempty and compact in Y,
if A is a closed set and -uniformly (w.r.t. ) attracting B, then .
- (ii)translation identity:
Definition 4.5 
The set is said to be kernel section at time , .
Definition 4.6 
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.
Lemma 4.2 
Under Assumptions 1 and 2 with , which is a weakly continuous semigroup, if acting on X is -uniformly (w.r.t. ) asymptotically compact, then it possesses an -uniform (w.r.t. ) attractor , which is compact in Y and attracts all the bounded subsets of X in the topology of Y.
where is a bounded neighborhood of the compact -uniformly attracting set in Y; 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 .
Lemma 4.3 
for any , , .
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 .
Lemma 4.5 
for all , ,
the translation group is weakly continuous on ,
is weakly compact.
Theorem 4.1 The family of processes corresponding to problem (1)-(5) is -weakly continuous and -weakly continuous.
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 . □
Lemma 4.6 
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).
for any , where , , , and are specified in (33), (41), (32) and (40), respectively.
From Theorem 4.2, the compactness of the Sobolev embedding , the compactness of the Sobolev trace embedding and Lemma 4.2, we have the following result.
where is the -uniformly (w.r.t. ) absorbing set in and is the section at of kernel of the process with symbol .
4.3 The existence of -uniform attractor
The main purpose of this subsection is to give an asymptotic a priori estimate for the unbounded part of the modular for the solution of problem (1)-(5) in the -norm.
where is the -uniformly (w.r.t. ) absorbing set and is the section at of kernel of the process with symbol