Abstract
In this paper, we are concerned with a system of nonlinear viscoelastic wave equations with initial and Dirichlet boundary conditions in (). Under suitable assumptions, we establish a general decay result by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system.
MSC: 35L05, 35L55, 35L70.
Keywords:
viscoelastic system; general decay; weak damping1 Introduction
In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave equations with weak damping
where () is a bounded domain with smooth boundary ∂Ω, u and v represent the transverse displacements of waves. The functions and denote the kernel of a memory, and are the nonlinearities.
In recent years, many mathematicians have paid their attention to the energy decay and dynamic systems of the nonlinear wave equations, hyperbolic systems and viscoelastic equations.
Firstly, we recall some results concerning single viscoelastic wave equation. Kafini and Tatar [1] considered the following Cauchy problem:
They established the polynomial decay of the firstorder energy of solutions for compactly supported initial data and for a not necessarily decreasing relaxation function. Later Tatar [2] studied the problem (1.2) with the Dirichlet boundary condition and showed that the decay of solutions was an arbitrary decay not necessarily at exponential or polynomial rate. Cavalcanti et al.[3] studied the following equation with Dirichlet boundary condition:
The authors established a global existence result for and an exponential decay of energy for . They studied the interaction within the and the memory term . Later on, several other results were published based on [46]. For more results on a single viscoelastic equation, we can refer to [714].
For a coupled system, Agre and Rammaha [15] investigated the following system:
where () is a bounded domain with smooth boundary. They considered the following assumptions on ():
(A_{1}) Let
(A_{2}) There exist two positive constants , such that for all , satisfies
Under the assumptions (A_{1})(A_{2}), they established the global existence of weak solutions and the global existence of small weak solutions with initial and Dirichlet boundary conditions. Moreover, they also obtained the blow up of weak solutions. Mustafa [16] studied the following system:
in with initial and Dirichlet boundary conditions, proved the existence and uniqueness to the system by using the classical FaedoGalerkin method and established a stability result by multiplier techniques. But the author considered the following different assumptions on () from (A_{1})(A_{2}):
() () are functions and there exists a function F such that
for all , where the constant and , for .
Han and Wang [17] considered the following coupled nonlinear viscoelastic wave equations with weak damping:
where is a bounded domain with smooth boundary ∂Ω. Under the assumptions (A_{1})(A_{2}) on (), the initial data and the parameters in the equations, they established the local existence, global existence uniqueness and finite time blow up properties. When the weak damping terms , were replaced by the strong damping terms , , Liang and Gao [18] showed that under certain assumption on initial data in the stable set, the decay rate of the solution energy is exponential when they take
and if , if . Moreover, they obtained that the solutions with positive initial energy blow up in a finite time for certain initial data in the unstable set. For more results on coupled viscoelastic equations, we can refer to [1921].
If we take in (1.4), the system will be transformed into (1.1). To the best of our knowledge, there is no result on general energy decay for the viscoelastic problem (1.1). Motivated by [16,17], in this paper, we shall establish the general energy decay for the problem (1.1) by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system. The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statement and the proof of our main result will be given in Section 3.
For the reader’s convenience, we denote the norm and the scalar product in by and , respectively. denotes a general constant, which may be different in different estimates.
2 Preliminaries and main result
To state our main result, in addition to (A_{1})(A_{2}), we need the following assumption.
(A_{3}) , , are differentiable functions such that
and there exist nonincreasing functions satisfying
Now, we define the energy functional
and the functional
where
The existence of a global solution to the system (1.1) is established in [17] as follows.
Proposition[17]
Let (A_{1})(A_{3}) hold. Assume that, and that, , whereis a computable constant and. Then the problem (1.1) has a unique global solutionsatisfying
We are now ready to state our main result.
Theorem 2.1Let (A_{1})(A_{3}) hold. Assume that, and that, , whereis a computable constant and. Then there exist constantssuch that, fortlarge, the solution of (1.1) satisfies
where
3 Proof of Theorem 2.1
In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate several lemmas.
Lemma 3.1Let, be the solution of (1.1). Then the following energy estimate holds for any:
Proof Multiplying the first equation of (1.1) by and the second equation by , respectively, integrating the results over Ω, performing integration by parts and noting that , we can easily get (3.1). The proof is complete. □
Lemma 3.2Under the assumption (A_{3}), the following hold:
Proof Using Hölder’s inequality, we get
On the other hand, we repeat the above proof with , instead of g, we can get (3.3). The proof is now complete. □
Lemma 3.3Let (A_{1})(A_{3}) hold and, be the solution of (1.1). Then the functionaldefined by
satisfies
Proof By (1.1), a direct differentiation gives
From the assumptions (A_{1})(A_{2}), we derive
and
By Young’s inequality and (3.2), we deduce for any
Similarly, we have
Using Young’s inequality and Poincaré’s inequality, we obtain for any
where λ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Similarly,
which together with (3.5)(3.9) gives
which together with (3.10) gives (3.4). The proof is complete. □
Lemma 3.4Let (A_{1})(A_{3}) hold and, be the solution of (1.1). Then the functionaldefined by
with
satisfies
Proof A direct differentiation for yields
Using the first equation of (1.1) and integrating by parts, we obtain
From Young’s inequality, Poincaré’s inequality and Lemma 3.2, we derive
Now, we estimate the first term on the righthand side of (3.17). Using the assumptions (A_{1})(A_{2}) and Young’s inequality, we arrive at
where we used the embedding for if or if and the fact proved in Lemma 5.1 in [17]. Combining (3.13)(3.18), we get
The same estimate to , we can derive
which together with (3.19) gives (3.11). The proof is now complete. □
Proof of Theorem 2.1 For , we define the functional by
and let
Using Lemma 3.1 and Lemmas 3.33.4, a direct differentiation gives
Now, we choose and , large enough so that
Inserting (3.21)(3.23) into (3.20), we have
Therefore, for two positive constants ω and C, we obtain
On the other hand, we choose even larger so that is equivalent to , i.e.,
Multiplying (3.25) by and using (A_{3}), we get
By virtue of (A_{3}) and , we have
Using (3.26), we can easily get
which together with (3.28) yields, for some positive constant η,
Integrating (3.30) over , we arrive at
which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper is a joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
Baowei Feng was supported by the Doctoral Innovational Fund of Donghua University with contract number BC201138, and Yuming Qin was supported by NNSF of China with contract numbers 11031003 and 11271066 and the grant of Shanghai Education Commission (No. 13ZZ048).
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