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Well-posedness for a class of wave equation with past history and a delay

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Abstract

In this paper, we consider a class of wave equation with past history and a delay term in the internal feedback. Namely, we investigate the following equation

$$u_{tt} - \alpha\triangle u + \int\limits_{-\infty}^{t} \mu(t - s)\triangle u(s){\rm d}s+\mu_1u_t + \mu_2u_t(x, t - \tau) + f(u) = h,$$

together with some suitable initial data and boundary conditions. The problem was considered by several authors, with \({\mu_2 = 0}\). The project of the present paper is to provide the well-posedness for this problem in a general setting which includes a delay term (\({\mu_2 \neq 0}\)). We also establish the exponential stability result when \({h(x) = 0}\).

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, People’s Republic of China

    Gongwei Liu & Hongwei Zhang

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  1. Gongwei Liu
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  2. Hongwei Zhang
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Correspondence to Gongwei Liu.

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Liu, G., Zhang, H. Well-posedness for a class of wave equation with past history and a delay. Z. Angew. Math. Phys. 67, 6 (2016). https://doi.org/10.1007/s00033-015-0593-z

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  • DOI: https://doi.org/10.1007/s00033-015-0593-z

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