Department of Applied Mathematics and Statistics

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.