Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Abstract

In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms

$A(t)\left|u_{t}\right|^{m-2} u_{t}-\Delta u+\int_{0}^{t} g(t-s) \Delta u(x, s) \mathrm{d} s=|u|^{p-2} u \log |u|.$

Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)\left|u_{t}\right|^{m-2} u_{t}$. To bypass this difficulty, we build up inverse Hölder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time. In addition, we can also give a decay rate under a general assumption on the relaxation functions satisfying $g^{\prime} \leq-\xi(t) H(g(t))$, $H(t)=t^{v}$, $t \geq 0, v>1.$ This improves the existing results.

Keywords

Viscoelastic term / blow up / decay estimate