Threshold, subthreshold and global unbounded solutions of superlinear heat equations
Analysis of PDEs
2025-10-28 v4
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 and decay to zero 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 strongly improve known results even in the model case of power nonlinearities, especially in the Sobolev critical and supercritical cases.
Keywords
Cite
@article{arxiv.2504.11813,
title = {Threshold, subthreshold and global unbounded solutions of superlinear heat equations},
author = {Pavol Quittner and Philippe Souplet},
journal= {arXiv preprint arXiv:2504.11813},
year = {2025}
}