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

DOI:
https://doi.org/10.1090/S0002-9947-04-03811-5

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 in a smooth bounded domain with zero Dirichlet boundary condition and initial condition , . In some cases we are also able to treat . 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:
https://doi.org/10.1090/S0002-9947-04-03811-5

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