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Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

  • Qiuyi Dai
  • Zhifeng YangEmail author
Article

Abstract

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation
$$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$
together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers  μ 1 and μ 2, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ 1 and μ 2. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ 1 = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).

Mathematics Subject Classification (2010)

35L05 35L20 35L70 93D15 

Keywords

Viscoelastic wave equation Global existence Energy decay Interior feedback 

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© Springer Basel 2013

Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceHunan Normal UniversityHunanPeople’s Republic of China
  2. 2.Department of Mathematics and Computational ScienceHengyang Normal UniversityHunanPeople’s Republic of China

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