Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 11.4.2019 at 14:00 Lecture room M-223

Pavol Quittner (KAMŠ FMFI UK):

Entire solutions of a semilinear heat equation

Abstract:

We characterize some classes of positive entire solutions of the nonlinear heat equation $u_t=\Delta u+u^p$,

where $p$ is supercritical in the Sobolev sense. In particular, if $p>p_L$ where $p_L$ denotes the Lepin exponent,

then any positive entire radial solution has to be a steady state (the condition $p\geq p_L$ is known to be necessary for such statement).

As an application, we show the convergence of rescaled profiles of solutions with type II blow-up and of global solutions with slow time decay.

This is a joint work with Peter Poláčik (University of Minnesota).