Seminar on Qualitative Theory of Differential Equations - Pavol Quittner (10.4.2025)
Thursday 10.4.2025 at 14:00, Lecture room M 223
Pavol Quittner:
Threshold, subthreshold and global unbounded solutions of superlinear heat equations
Abstract:
We consider the semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions, lying on or below the boundary between blow-up and global existence, respectively. For the Cauchy-Dirichlet problem, we prove the boundedness of any subthreshold solution. This implies, in particular, that all global unbounded solutions - if they exist - are threshold solutions. For the Cauchy problem, these properties fail in general but we show that they become true for a suitably modified notion of threshold. Our results improve known results even in the special case of power nonlinearities, especially in the Sobolev critical and supercritical cases.
This is a joint work with Philippe Souplet.