Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 85, pp. 1-15.

Title: Stability of initial-boundary value problem for quasilinear viscoelastic equations
       
Authors: Kun-Peng Jin (Chongqing Univ. of Posts and Telecom., Chongqing, China)
         Jin Liang (Shanghai Jiao Tong Univ., Shanghai, China)
         Ti-Jun Xiao (Fudan Univ., Shanghai 200433, China)

Abstract:
 We investigate the stability of the initial-boundary value problem for the
 quasilinear viscoelastic equation
 $$\displaylines{
 |u_t|^{\rho}u_{tt}-\Delta u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds=0,
 \quad \text{in }\Omega\times(0,+\infty),\cr
 u=0,\quad  \text{in }\partial\Omega\times(0,+\infty),\cr
 u(\cdot, 0)=u_0(x),\quad u_t(\cdot, 0)=u_1(x), \quad \text{in }\Omega,
 }$$
 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.