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1997 Self-similar blow-up for a quasilinear parabolic equation with gradient diffusion and exponential source
Chris J. Budd, James W. Dold, Victor A. Galaktionov
Adv. Differential Equations 2(1): 85-124 (1997).

Abstract

We study the asymptotic behaviour near a finite blow-up time $t=T$ of the solutions to the initial-boundary value problem for the quasilinear equation $$ u_t = \nabla \cdot (|\nabla u|^\sigma \nabla u) + e^u \mbox{ in } \{ | x| < R\} \times (0,T), \ \ \sigma >0, $$ with zero Dirichlet boundary condition and a radial symmetric initial function $u_0 (|x|) >0$ in $\{ |x| <R\}, \ u'_0 (r)<0$ in $(0,R)$. We prove single point blow-up and different sharp lower and upper estimates of the solution as $t \rightarrow T^-$ and of the final-time profile. For the one-dimensional problem the asymptotic behaviour is proved to be described by nonconstant self-similar solutions of the form $u_* (x,t) =-\log (T-t) + \theta (\xi), \ \xi = x/(T-t)^{1/(\sigma+2)}$, where $\theta (\xi) \sim - (\sigma+2) \log \xi$ as $\xi \rightarrow\infty$.

Citation

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Chris J. Budd. James W. Dold. Victor A. Galaktionov. "Self-similar blow-up for a quasilinear parabolic equation with gradient diffusion and exponential source." Adv. Differential Equations 2 (1) 85 - 124, 1997.

Information

Published: 1997
First available in Project Euclid: 24 April 2013

zbMATH: 1023.35516
MathSciNet: MR1424764

Subjects:
Primary: 35K60
Secondary: 35B05

Rights: Copyright © 1997 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.2 • No. 1 • 1997
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