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July/August 2009 Pointwise decay for the solutions of degenerate and singular parabolic equations
Petri Juutinen, Peter Lindqvist
Adv. Differential Equations 14(7/8): 663-684 (July/August 2009).

Abstract

We study the asymptotic behavior, as $t\to\infty$, of the solutions to the evolutionary $p$-Laplace equation \[ v_t={\operatorname{div}}( | {\nabla v} | ^{p-2}\nabla v), \] with time-independent lateral boundary values. We obtain the sharp decay rate of $\max_{x\in{\Omega}} | {v(x,t)-u(x)}| $, where $u$ is the stationary solution, both in the degenerate case $p > 2$ and in the singular case $1 < p > 2$. A key tool in the proofs is the Moser iteration, which is applied to the difference $v(x,t)-u(x)$. In the singular case, we construct an example proving that the celebrated phenomenon of finite extinction time, valid for $v(x,t)$ when $u\equiv 0$, does not have a counterpart for $v(x,t)-u(x)$.

Citation

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Petri Juutinen. Peter Lindqvist. "Pointwise decay for the solutions of degenerate and singular parabolic equations." Adv. Differential Equations 14 (7/8) 663 - 684, July/August 2009.

Information

Published: July/August 2009
First available in Project Euclid: 18 December 2012

zbMATH: 1182.35036
MathSciNet: MR2527689

Subjects:
Primary: 35B40, 35K55, 35K65

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.14 • No. 7/8 • July/August 2009
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