Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay |
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Authors: | Qiuyi Dai Zhifeng Yang |
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Institution: | 1. College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China 2. Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421008, Hunan, People’s Republic of China
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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). |
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