Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux

Zhiqing  Liu; Zhong Bo  Fang

doi:10.3934/dcdsb.2016113

Article Contents

2016, Volume 21, Issue 10: 3619-3635. Doi: 10.3934/dcdsb.2016113

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Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux

Zhiqing Liu^1, and
Zhong Bo Fang^1,

1.
School of Mathematical Sciences, Ocean University of China, Qingdao 266100

Received: December 31, 2015

Revised: March 31, 2016

Published: November 2016

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Abstract

This paper deals with blow-up phenomena for an initial boundary value problem of a nonlocal quasilinear parabolic equation with time-dependent coefficients in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficients and nonlinearities to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, upper and lower bounds of $t^{\ast}$ are obtained under suitable measure in high-dimensional spaces. Finally, some application examples are presented.

Keywords:

Nonlocal quasilinear parabolic equation,
time-dependent coefficients,
blow-up time,
upper bound,
lower bound.

Mathematics Subject Classification: Primary: 35K59, 35B44; Secondary: 35B40.

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