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Infinite time blow-up for superlinear parabolic problems with localized reaction


Author: Philippe Souplet
Journal: Proc. Amer. Math. Soc. 133 (2005), 431-436
MSC (2000): Primary 35K60, 35B40
DOI: https://doi.org/10.1090/S0002-9939-04-07707-X
Published electronically: September 16, 2004
MathSciNet review: 2093064
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Abstract: We consider the nonlocal diffusion equation

\begin{displaymath}u_t-u_{xx}=u^p(t,x_0(t)),\end{displaymath}

on the space interval $(0,1)$, with Dirichlet boundary conditions. It is known that if the curve $x_0(t)$ remains in a compact subset of $(0,1)$ for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on $x_0$ is sharp: for a large class of functions $x_0(t)$approaching the boundary as $t\to\infty$, blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.


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Additional Information

Philippe Souplet
Affiliation: Département de Mathématiques, INSSET Université de Picardie, 02109 St-Quentin, France – and – Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles, 45 avenue des États-Unis, 78035 Versailles, France
Email: souplet@math.uvsq.fr

DOI: https://doi.org/10.1090/S0002-9939-04-07707-X
Keywords: Semilinear diffusion equation, localized reaction, nonlocal parabolic problem, blow-up in infinite time, asymptotic behavior
Received by editor(s): December 4, 2002
Published electronically: September 16, 2004
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2004 American Mathematical Society

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