(untitled) (Journal of Partial Differential Equations, 2024)
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)|u_t|^m-2u_t-Δ u+_0^tg(t-s)Δ u(x,s)ds=|u|^p-2u|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)|u_t|^m-2u_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'≤ -ξ(t)H(g(t)), H(t)=t^ u, t≥ 0, u>1. This improves the existing results.