TY - JOUR
T1 - Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem
AU - Li , Fang
AU - Zhang , Jingjing
JO - Journal of Partial Differential Equations
VL - 3
SP - 263
EP - 277
PY - 2024
DA - 2024/08
SN - 37
DO - http://doi.org/10.4208/jpde.v37.n3.3
UR - https://global-sci.org/intro/article_detail/jpde/23342.html
KW - Viscoelastic term, blow up, decay estimate.
AB - <div style="text-align: justify;">In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms $$A(t)|u_{t}|^{m-2}u_{t}-\Delta u+\int_0^{t}g(t-s)\Delta u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)|u_{t}|^{m-2}u_{t}$. 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 $g&#39;\leq -\xi(t)H(g(t)),~H(t)=t^\nu,~t\geq 0,~\nu&gt;1$. This improves the existing results.</div>