Seminar on Qualitative Theory of Differential Equations - Pavol Quittner (1.10.2020)
Thursday 1.10.2020 at 14:00, Lecture room C
By: Pavol Quittner
An optimal Liouville theorem for a semilinear heat equation
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.