Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem (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 $$A(t)|u_{t}|^{m-2}u_{t}-Δ u+\int_0^{t}g(t-s)Δ u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ 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 $g'≤ -ξ(t)H(g(t)),~H(t)=t^ u,~t≥ 0,~ u>1$. This improves the existing results.
Keywords: Viscoelastic term; blow up; decay estimate.