Threshold, subthreshold and global unbounded solutions of superlinear heat equations

TL;DR

The paper addresses threshold, subthreshold, and global unbounded solutions for the parabolic equation $u_t-\Delta u=f(u)$ with a superlinear nonlinearity $f$, treating both convex and nonconvex nonlinearities across bounded domains and $\mathbb{R}^n$. It develops a toolkit based on energy methods, Kaplan’s eigenfunction technique, and smoothing estimates to establish boundedness and decay of subthresholds, and to classify global unbounded solutions as (modified) thresholds in various geometric settings, with sharp results in the Sobolev-critical regime ($p_S$) and beyond ($p_{JL}$). The results extend and strengthen known phenomena for $f(u)\sim u^p$, including nonconvex growth such as $f(u)=u^p\log^a(2+u^2)$ or $e^u-u-1$, by providing conditions under which subthresholds remain bounded or decay and by clarifying when global unbounded solutions are thresholds or $\varepsilon f$-thresholds. The authors combine subsolution methods (e.g., $u+\varepsilon f(u)$ or $(1+\varepsilon)u$), time-decay/energy arguments, and geometric scaling/intersection properties of entire solutions to $-\Delta U=U^{p_S}$ to produce a unified, sharp theory of threshold behavior for superlinear parabolic equations. These results have implications for long-time dynamics and blow-up, and substantially broaden the threshold framework beyond convex, subcritical settings to critical and supercritical nonlinearities on varied domains.

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.

Theorems & Definitions (32)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof