General decay for a system of nonlinear viscoelastic wave equations with weak damping
© Feng et al.; licensee Springer 2012
Received: 19 August 2012
Accepted: 26 November 2012
Published: 13 December 2012
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.
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.
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 [4–6]. For more results on a single viscoelastic equation, we can refer to [7–14].
where () is a bounded domain with smooth boundary. They considered the following assumptions on ():
with , if and if ; .
in with initial and Dirichlet boundary conditions, proved the existence and uniqueness to the system by using the classical Faedo-Galerkin method and established a stability result by multiplier techniques. But the author considered the following different assumptions on () from (A1)-(A2):
for all , where the constant and , for .
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 [19–21].
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 (A1)-(A2), we need the following assumption.
The existence of a global solution to the system (1.1) is established in  as follows.
We are now ready to state our main result.
3 Proof of Theorem 2.1
In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate several lemmas.
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. □
On the other hand, we repeat the above proof with , instead of g, we can get (3.3). The proof is now complete. □
for all .
which together with (3.10) gives (3.4). The proof is complete. □
which together with (3.19) gives (3.11). The proof is now complete. □
for some fixed .
which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □
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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