Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary

Authors: Hamid Ghidouche, Philippe Souplet and Domingo Tarzia
Journal: Proc. Amer. Math. Soc. 129 (2001), 781-792
MSC (2000): Primary 35K55, 35R35, 80A22, 35B35, 35B40
Published electronically: September 20, 2000
MathSciNet review: 1802001
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We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in $L^\infty$ norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).

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Additional Information

Hamid Ghidouche
Affiliation: Laboratoire Analyse, Géométrie et Applications, UMR CNRS 7539, Institute Galilée, Université Paris Nord, 93430 Villetaneuse, France

Philippe Souplet
Affiliation: Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin, France and Laboratoire de Mathématiques Appliquées, UMR CNRS 7641, Université de Versailles 45 avenue des Etats-Unis, 78302 Versailles, France

Domingo Tarzia
Affiliation: Departamento de Matemática, FCE, Universidad Austral, Paraguay 1950, 2000, Rosario, Argentina

Keywords: Nonlinear reaction-diffusion equation, free boundary condition, Stefan problem, global existence, boundedness of solutions, decay, stability, finite time blow up.
Received by editor(s): May 11, 1999
Published electronically: September 20, 2000
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society