Existence and asymptotic stability of a viscoelastic wave equation with a delay
Existence and asymptotic stability of a viscoelastic wave equation with a delay
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DOI：
10.1007/s00033-011-0145-0
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发表时间：
2011-07
期刊：
影响因子：
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通讯作者：
M. Kirane;B. Said-Houari
中科院分区：
文献类型：
--
作者：
M. Kirane;B. Said-Houari
In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem $$u_{tt}(x,t)-\Delta u(x,t)+\int\limits_{0}^{t}g(t-s){\Delta}u(x,s){d}s+\mu_{1}u_{t}(x,t)+\mu_{2} u_{t}(x,t-\tau)=0$$together with initial conditions and boundary conditions of Dirichlet type. Hereis a positive real valued decreasing function andμ1,μ2are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo–Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals.
In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem $$u_{tt}(x,t)-\Delta u(x,t)+\int\limits_{0}^{t}g(t-s){\Delta}u(x,s){d}s+\mu_{1}u_{t}(x,t)+\mu_{2} u_{t}(x,t-\tau)=0$$together with initial conditions and boundary conditions of Dirichlet type. Hereis a positive real valued decreasing function andμ1,μ2are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo–Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals.