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 damping
In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave equations with weak damping
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  considered the following Cauchy problem:
They established the polynomial decay of the first-order energy of solutions for compactly supported initial data and for a not necessarily decreasing relaxation function. Later Tatar  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. 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 [4-6]. For more results on a single viscoelastic equation, we can refer to [7-14].
For a coupled system, Agre and Rammaha  investigated the following system:
Under the assumptions (A1)-(A2), 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  studied the following system:
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):
Han and Wang  considered the following coupled nonlinear viscoelastic wave equations with weak damping:
where is a bounded domain with smooth boundary ∂Ω. Under the assumptions (A1)-(A2) 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  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 [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.
2 Preliminaries and main result
To state our main result, in addition to (A1)-(A2), we need the following assumption.
Now, we define the energy functional
and the functional
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. □
Lemma 3.2Under the assumption (A3), the following hold:
Proof Using Hölder’s inequality, we get
Proof By (1.1), a direct differentiation gives
From the assumptions (A1)-(A2), we derive
Similarly, we have
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. □
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 right-hand side of (3.17). Using the assumptions (A1)-(A2) 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 . Combining (3.13)-(3.18), we get
which together with (3.19) gives (3.11). The proof is now complete. □
Using Lemma 3.1 and Lemmas 3.3-3.4, a direct differentiation gives
Inserting (3.21)-(3.23) into (3.20), we have
Therefore, for two positive constants ω and C, we obtain
Using (3.26), we can easily get
which together with (3.28) yields, for some positive constant η,
which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □
The authors declare that they have no competing interests.
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.
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).
Cavalcanti, MM, Domingos Cavalcanti, VN, Ferreira, J: Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math. Methods Appl. Sci.. 24, 1043–1053 (2001). Publisher Full Text
Cavalcanti, MM, Domingos Cavalcanti, VN, Prates Filho, JS, Soriano, JA: Existence and uniform decay rates for viscoelastic problems with non-linear boundary damping. Differ. Integral Equ.. 14(1), 85–116 (2001)
Messaoudi, SA, Tatar, N-e: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci.. 30, 665–680 (2007). Publisher Full Text
Messaoudi, SA, Tatar, N-e: Exponential and polynomial decay for a quasilinear viscoelastic problem. Math. Nachr.. 282, 1443–1450 (2009). Publisher Full Text
Fabrizio, M, Polidoro, S: Asymptotic decay for some differential systems with fading memory. Appl. Anal.. 81(6), 1245–1264 (2002). Publisher Full Text
Messaoudi, SA: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl.. 341, 1457–1467 (2008). Publisher Full Text
Messaoudi, SA: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. TMA. 69, 2589–2598 (2008). Publisher Full Text
Messaoudi, SA, Fareh, A: General decay for a porous thermoelastic system with memory: the case of equal speeds. Nonlinear Anal. TMA. 74, 6895–6906 (2011). Publisher Full Text
Tatar, N-e: Exponential decay for a viscoelastic problem with a singular problem. Z. Angew. Math. Phys.. 60(4), 640–650 (2009). Publisher Full Text
Mediden, M, Tatar, N-e: Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl. Math. Comput.. 167(2), 1221–1235 (2005). Publisher Full Text
Mustafa, MI: Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal.: Real World Appl.. 13, 452–463 (2012). Publisher Full Text
Han, X, Wang, M: Global existence and blow up of solutions for a system of nonlinear viscoelastic wave equations with damping and source. Nonlinear Anal. TMA. 71, 5427–5450 (2009). Publisher Full Text
Liu, W: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. TMA. 71, 2257–2267 (2009). Publisher Full Text
Zhang, J: On the standing wave in coupled nonlinear Klein-Gordon equations. Math. Methods Appl. Sci.. 26, 11–25 (2003). Publisher Full Text