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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence and asymptotic behavior for a singular parabolic equation


Authors: Juan Dávila and Marcelo Montenegro
Journal: Trans. Amer. Math. Soc. 357 (2005), 1801-1828
MSC (2000): Primary 35B40, 35K55
Published electronically: December 29, 2004
MathSciNet review: 2115077
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Abstract: We prove global existence of nonnegative solutions to the singular parabolic equation $u_t -\Delta u + \raise 1.5pt\hbox{$\chi$ }_{ \{ u>0 \} } ( -u^{-\beta} + \lambda f(u) )=0$ in a smooth bounded domain $\Omega\subset\mathbb{R} ^N$ with zero Dirichlet boundary condition and initial condition $u_0 \in C(\Omega)$, $u_0 \geq 0$. In some cases we are also able to treat $u_0 \in L^\infty(\Omega)$. Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called ``quenching''. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.


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Additional Information

Juan Dávila
Affiliation: Departamento de Ingeniería Matemática, CMM (UMR CNRS), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
Email: jdavila@dim.uchile.cl

Marcelo Montenegro
Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Caixa Postal 6065, CEP 13084-970, Campinas, SP, Brasil
Email: msm@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03811-5
PII: S 0002-9947(04)03811-5
Keywords: Singular parabolic equation, quenching problem
Received by editor(s): July 18, 2003
Published electronically: December 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society