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

Forward self-similar solution with a moving singularity for a semilinear parabolic equation

Pages: 313 - 331, Volume 26, Issue 1, January 2010      doi:10.3934/dcds.2010.26.313

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Shota Sato - Mathematical Institute, Tohoku University, Sendai 980-8578, Japan (email)
Eiji Yanagida - Mathematical Institute Tohoku University, 6-3Aoba, Aramaki, Aoba-ku, Sendai-shi, 980-8578, Japan (email)

Abstract: We study the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a time-local solution with prescribed moving singularities. Our concern in this paper is the existence of a time-global solution. By using a perturbed Haraux-Weissler equation, it is shown that there exists a forward self-similar solution with a moving singularity. Using this result, we also obtain a sufficient condition for the global existence of solutions with a moving singularity.

Keywords:  Semilinear parabolic equation, forward self-similar, moving singularity, critical exponent.
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B33.

Received: January 2009;      Revised: June 2009;      Available Online: October 2009.