# American Institute of Mathematical Sciences

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November  2012, 32(11): 4027-4043. doi: 10.3934/dcds.2012.32.4027

## Asymptotic behavior of singular solutions for a semilinear parabolic equation

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

Received  May 2011 Revised  August 2011 Published  June 2012

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.
Citation: Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
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