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arXiv · 2504.11813

Threshold, subthreshold and global unbounded solutions of superlinear heat equations

Pavol Quittner·Philippe Souplet
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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.

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Pavol Quittner, Philippe Souplet. 2025-10-27. Threshold, subthreshold and global unbounded solutions of superlinear heat equations. https://arxiv.org/abs/2504.11813

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