A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

✍ Scribed by Peter Poláčik; Pavol Quittner

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 173 KB
• Volume: 64
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.07.016

✦ Synopsis

We consider the semilinear parabolic equation u t = u + u p on R N , where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x ∈ R N and t ∈ R. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t 0, then it necessarily converges to 0, as t → ∞, uniformly with respect to x ∈ R N .