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Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay

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Abstract

In this paper, we consider initial-boundary value problem of Euler–Bernoulli viscoelastic equation with a delay term in the internal feedbacks. Namely, we study the following equation

$$u_{tt}(x,t)+ \Delta^2 u(x,t)-\int\limits_0^t g(t-s)\Delta^2 u(x,s){\rm d}s+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=0 $$

together with some suitable initial data and boundary conditions in \({\Omega\times (0,+\infty)}\) . For arbitrary real numbers μ 1 and μ 2, we prove that the above-mentioned model has a unique global solution under suitable assumptions on the relaxation function g. Moreover, under some restrictions on μ 1 and μ 2, exponential decay results of the energy for the concerned problem are obtained via an appropriate Lyapunov function.

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Authors and Affiliations

  1. Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421002, Hunan People’s Republic of China

    Zhifeng Yang

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  1. Zhifeng Yang
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Correspondence to Zhifeng Yang.

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Project supported by the Natural Science Foundation of Hunan Province, China (Grant No. 14JJ7070) and the Key Built Disciplines of Hunan Province (No. [2011]76).

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Yang, Z. Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay. Z. Angew. Math. Phys. 66, 727–745 (2015). https://doi.org/10.1007/s00033-014-0429-2

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  • DOI: https://doi.org/10.1007/s00033-014-0429-2

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