Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 1.10.2020 at 14:00 Lecture room C

Pavol Quittner (KAMŠ FMFI UK):

An optimal Liouville theorem for a semilinear heat equation

Abstract:

Liouville-type theorems for entire solutions of scaling invariant nonlinear

parabolic equations and systems guarantee optimal universal estimates

of solutions of related initial and initial-boundary value problems,

including estimates of their singularities and decay.

In this talk I will first review known Liouville-type theorems

for a semilinear heat equation (sometimes called the Fujita equation)

and then I will give a sketch of the proof of a Liouville-type theorem

guaranteeing the nonexistence of positive entire solutions

of the Fujita equation in the full subcritical range.

Our approach can also be used for a class of semilinear parabolic systems

and the linear heat equation with a nonlinear boundary condition.