Work

Threshold, subthreshold, and global unbounded solutions of superlinear heat equations

Year: 2025

Type: article

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.

Source: Proceedings of the London Mathematical Society

DOI: 10.1112/plms.70110

Authors: Pavol Quittner, Philippe Souplet

Institutions: Comenius University Bratislava, Université Sorbonne Paris Nord, Laboratoire Analyse, Géométrie et Applications

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Agentúra na Podporu Výskumu a Vývoja, Vedecká Grantová Agentúra MŠVVaŠ SR a SAV

Awards: G1149588761, Kvalitatívne vlastnosti evolučných úloh z prírodných a technických vied, G8088548421, Asymptotické vlastnosti riešení eliptických a parabolických úloh