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

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem
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Year:    2024

Author:    Fang Li, Jingjing Zhang

Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 3 : pp. 263–277

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 $$A(t)|u_{t}|^{m-2}u_{t}-\Delta u+\int_0^{t}g(t-s)\Delta u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ 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 $g'\leq -\xi(t)H(g(t)),~H(t)=t^\nu,~t\geq 0,~\nu>1$. This improves the existing results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v37.n3.3

Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 3 : pp. 263–277

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    15

Keywords:    Viscoelastic term blow up decay estimate.

Author Details

Fang Li

Jingjing Zhang