Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Singular backward self-similar solutions of a semilinear parabolic equation

Pages: 897 - 906, Volume 4, Issue 4, August 2011      doi:10.3934/dcdss.2011.4.897

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Shota Sato - Mathematical Institute, Tohoku University, Sendai 980-8578, Japan (email)
Eiji Yanagida - Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan (email)

Abstract: We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.

Keywords:  Semilinear parabolic equation, backward self-similar solution, singular solution, critical exponent.
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B33.

Received: September 2009;      Revised: December 2009;      Available Online: November 2010.