Electron. J. Differential Equations, Vol. 2020 (2020), No. 85, pp. 1-15.

Stability of initial-boundary value problem for quasilinear viscoelastic equations

Kun-Peng Jin, Jin Liang, Ti-Jun Xiao

We investigate the stability of the initial-boundary value problem for the quasilinear viscoelastic equation

where $\Omega$ is a bounded domain of $\mathbb{R}^{n}\; (n\geq 1)$ with smooth boundary $\partial\Omega$, $\rho$ is a positive real number, and g(t) is the relaxation function. We present a general polynomial decay result under some weak conditions on g, which generalizes and improves the existing related results. Moreover, under the condition $g'(t)\leq -\xi(t)g^{p}(t)$, we obtain uniform exponential and polynomial decay rates for $1\leq p<2$, while in the previous literature only the case $1\leq p<3/2$ was studied. Finally, under a general condition $g'(t)\leq -H(g(t))$, we establish a fine decay estimate, which is stronger than the previous results.

Submitted November 11, 2019. Published July 30, 2020.
Math Subject Classifications: 35Q74, 35B35, 74H55, 74H40, 93D15.
Key Words: Quasilinear viscoelastic equation; polynomial and exponential decay; relaxation function; uniform decay.

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Kun-Peng Jin
School of Science
Chongqing University of Posts and Telecommunications
Chongqing 400065, China
email: kjin11@fudan.edu.cn
Jin Liang
School of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai 200240, China
email: jinliang@sjtu.edu.cn
Ti-Jun Xiao
Shanghai Key Laboratory for Contemporary Applied Mathematics
School of Mathematical Sciences
Fudan University
Shanghai 200433, China
email: tjxiao@fudan.edu.cn

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