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Regularity of the attractor for strongly damped wave equations with nonlinearity
Journal of Inequalities and Applications volume 2014, Article number: 396 (2014)
The long time behavior of the solutions for the strongly damped wave equation is considered with nonlinear damping, a nonlinear forcing term, and with a periodic boundary condition. We prove that the global attractor which captures all trajectories in is a compact set in .
MSC:35B40, 34A35, 37L30.
For , we consider the strongly damped wave equation on a bounded domain Ω in with smooth boundary:
supplemented with the periodic boundary conditions and initial conditions
where the strongly damped coefficient α is a positive constant, the damped function is continuous, and the forcing term is a nonlinear term satisfying some growth conditions, is the external force.
When , (1.1) reduces to a usual wave equation with nonlinear damping, which arises as an evolutionary mathematical model in various systems (cf. [1, 2]), for example: (i) modeling a continuous Josephson junction with specific h, g and f; (ii) modeling a hybrid system of nonlinear waves and nerve conduct; and (iii) modeling a phenomenon in quantum mechanics and which has been studied widely by using of the concept of global attractors; see, for example, [2–10] for the linear damping case, and [11–18] for the nonlinear damping case. There are some results on the regularity in these papers; for example, in  the authors gave the dimensionality and related properties of the global attractor, and in , the authors presented a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation (when and ) with nonlinearity for the critical growth. But for , the case can be complex.
There is a large literature on the asymptotic behavior of solutions to (1.1), (1.2) (cf. [1, 19–23]). For the strongly damped wave equations,  had proved the uniform boundedness of the global attractor for very strong damping in and obtained an estimate of the upper bound of the Hausdorff dimension of an attractor for the strongly damped nonlinear wave equation (1.1). Furthermore, when , Pata and Zelik in  had proved the existence of compact global attractors of optimal regularity, i.e., the compact global attractor on is a bounded subset of . The aim of this paper is to prove that the dynamical system associated with (1.1) possesses a compact global attractor in , i.e., we will prove that the global attractor in is a compact set in . This implies that .
The paper is organized as follows. In Section 2, we present the existence, uniqueness, and continuous dependence of solutions for problem (1.1), and we establish the existence of an absorbing set in . In Section 3, we prove the existence of the global attractor in . In Section 4, we establish the regularity of the attractor, i.e., that the global attractor is a compact subset in .
We assume that , and , with
and the functions h, f satisfy the following conditions:
(1) The asymptotic sign condition(2.1)
(2) Let , there exist constants such that(2.2)
where denotes the absolute value of the number in ℝ. We have(2.3)
There exist two constants such that(2.4)
Let , then the system (1.1)-(1.2) is equivalent to the following initial value problem in :
where , ,
By the assumption (2.1)-(2.4), it is easy to check that the function is continuously differentiable and globally Lipschitz continuous with respect to Y. By the classical theory concerning the existence and uniqueness of the solutions of evolution differential equations (cf. ), we have the following lemma (see  for details).
Lemma 2.1 Consider the initial value problem (2.5) in . If (2.1)-(2.4) hold, then, for any , there exists a unique continuous function such that and satisfies the integral equation
is called the mild solution of (2.5), and is jointly continuous in t and , and
For any , we can introduce a map
where is the solution of (2.5), and then is a continuous semigroup on .
Consider the map defined as
For some , let
Obviously, is symmetric and positive definite, so we have the following Poincaré type inequality:
Let , then (1.1) can be written in the equivalent form
and by (2.4), satisfies
where , are two constants.
To construct an attractor for (1.1), we make the following assumptions. Let , , where k is chosen as
Equation (2.9) can be written as
We define a new weighted inner product and norm in as
for any , where μ is chosen as
Obviously, the norm in (2.13) is equivalent to the usual norm in E.
Lemma 2.2 (see , Lemma 1)
For any , if
Proposition 2.1 The semigroup possesses an absorbing set .
Proof Let be the solution of (2.12). Taking the inner product of (2.12) with φ, we have
By , we have
by (2.18) and (2.19), we have
Let . By (2.15) and (2.2), there exists such that
where . By (2.20) and (2.21),
Using the Gronwall lemma, we have
Note that (2.2) and (2.23) imply
For any bounded set B of E, where , if , there exists such that . Therefore, for the solution of (2.12) with ,
completes the proof. □
3 Existence of the global attractor in
Theorem 3.1 (see [, I.1.1])
Let be a continuous semigroup on that possesses an absorbing ball in . Let us assume that, for any ,
For every bounded set B, there existsthat depends on B, such thatis relatively compact in.
For every bounded set B,
Then possesses a global attractor that is compact in .
Remark 3.1 (see )
We characterize such an attractor as
Theorem 3.2 The semigroup possesses a global attractor in .
Proof We consider and such that
and we introduce the splitting where satisfies
and is the solution of
We now define the families of maps and in , where
First step: We prove that is bounded in . The system (3.2) can be written as
Similar to Proposition 2.1, we obtain
Now we multiply (3.2) by and integrate on Ω to obtain
Then (3.8) can be rewritten as
for the first term on the right-hand side of (3.9), we have
By (3.9)-(3.10), we have
Let , using the Gronwall lemma, we have
By (3.7), we have
Note that (3.2) implies that
Then we obtain
Proposition 2.1, (3.8), and (3.12) imply that is bounded in .
Second step: Let ; we will prove that there exists a independence on ϵ such that
Multiplying (3.3) by , we thus obtain
For the last term on the left-hand side of (3.13), by (2.3), Proposition 2.1, and (3.7), there exists ξ such that
By the Gronwall and Poincaré inequalities,
By (3.12), (3.15), and the following lemma we see that is compact in .
Lemma 3.1 (see )
Let X be a complete metric space and A be a subset in X, such that
with and is compact in X, then A is compact in X.
Third step: Let , by (3.4) and Lemma 2.2, we have
For the right-hand side of (3.16), by (2.1), we have
Let . Equations (3.16) and (3.17) lead to
In other words we have
uniformly in bounded sets. Then from Theorem 3.1, (3.18), and the compactness of , for the system (1.1) there exists a global attractor in . □
Remark 3.2 It is easy to see that the semigroup
defined by (2.12) has the following relation with :
where is an isomorphism of E:
Since the semigroup defined by (1.1) possesses a global attractor , by (3.19), also possesses a global attractor .
4 Regularity of the attractor
In this section, we suppose that and prove that the attractor is compact in . Let . We set
where is the solution of
We use Remark 3.1 to prove that . Let . Let and a sequence of a real numbers as , such that
We also have
We deduce from (3.12) and (3.20) that
From (4.6), we infer that there exist subsequences and , and such that
From (4.4), (4.5), and (4.7), we have
We conclude that and then
In the following, we use the famous argument of  to show that the attractor is actually a compact set in .
Theorem 4.1 The semigroup in possesses a global attractor which is compact subset of .
Proof Multiplying (1.1) by , we have
We put , , then we have
this shows that
We consider a sequence , and we may assume that, up to a subsequence,
We want to prove the strong convergence of in ; this will give the compactness of in .
For a given and up to subsequence extraction, we may assume that a.e. in ,
Using (4.13) for , we have
By the Lebesgue dominated convergence theorem, similar to (3.8)-(3.12), we have
passing to the limit sup in (4.14) and taking we have
The last term of (4.16) follows from (4.14) for and , and we have
We replace I in (4.16), we obtain
It is a standard matter to prove
It follows from (4.18) that
This completes the proof of lemma. □
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This work is supported by the National Natural Science Foundation of China (11101265) and Shanghai Education Research and Innovation Key Project (14ZZ157).
The author declares to have no competing interests.
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Li, H. Regularity of the attractor for strongly damped wave equations with nonlinearity. J Inequal Appl 2014, 396 (2014). https://doi.org/10.1186/1029-242X-2014-396
- global attractor
- strongly damped wave equation
- compact set