Stability of initial-boundary value problem for quasilinear viscoelastic equations

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.

Keywords

• quasilinear viscoelastic equation
• polynomial and exponential decay
• relaxation function
• uniform decay