Asymptotic behavior of singular solutions for a semilinear parabolicequation

Shota Sato; Eiji Yanagida

doi:10.3934/dcds.2012.32.4027

Article Contents

2012, Volume 32, Issue 11: 4027-4043. Doi: 10.3934/dcds.2012.32.4027

This issue Previous Article Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities Next Article Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity

Asymptotic behavior of singular solutions for a semilinear parabolic equation

Shota Sato^1, and
Eiji Yanagida^2,

1.
Mathematical Institute, Tohoku University, Sendai 980-8578
2.
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551

Received: May 2011

Revised: August 2011

Published: November 2012

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Abstract

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It is known that in some range of parameters, this equation has a family of singular steady states with ordered structure. Our concern in this paper is the existence of time-dependent singular solutions and their asymptotic behavior. In particular, we prove the convergence of solutions to singular steady states. The method of proofs is based on the analysis of a related linear parabolic equation with a singular coefficient and the comparison principle.

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Mathematics Subject Classification: Primary: 35K58; Secondary: 35B33, 35B35.

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