Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $R^n$

TL;DR

This paper establishes the boundedness and decay properties of global solutions to a supercritical semilinear heat equation in $\mathbb{R}^n$ without symmetry assumptions, using Morrey space techniques and energy estimates.

Contribution

It introduces a novel approach using Morrey space analysis for supercritical heat equations, simplifying previous methods and extending results to general domains.

Findings

Global solutions are bounded and decay faster than $t^{-1/(p-1)}$.

The set of initial data leading to global solutions is open.

Borderline solutions blow up then become classical and decay.

Abstract

We prove the boundedness of global classical solutions for the semilinear heat equation $u_t-\Delta u= |u|^{p-1}u$ in the whole space $R^n$, with $n\ge 3$ and supercritical power $p>(n+2)/(n-2)$. This is proved {\rmb without any radial symmetry or sign assumptions}, unlike in all the previously known results for the Cauchy problem, and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter-examples. Moreover, we show that any global classical solution has to decay in time faster than $t^{-1/(p-1)}$, which is also optimal and in contrast with the subcritical case. The proof relies on nontrivial modifications of techniques developed by Chou, Du and Zheng [Calc. Var. PDE 2007] and by Blatt and Struwe [IMRN, 2015] for the case of convex bounded domains. They are based on weighted energy estimates of Giga-Kohn type, combined with an analysis of the equation in a suitable Morrey space. We in particular simplify the approach of Blatt and Struwe by establishing and using a result on global existence and decay for small initial data in the critical Morrey space $M^{2,4/(p-1)}(R^n)$, rather than $\eps$-regularity in a parabolic Morrey space. This method actually works for any convex, bounded or unbounded, smooth domain, but at the same time captures some of the specific behaviors associated with the case of the whole space $R^n$. As a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies, and we show that the so-called "borderline" global weak solutions blow up in finite time and then become classical again and decay as $t\to\infty$. All these results put into light the key role played by the Morrey space $M^{2,4/(p-1)}$ in the understanding of the structure of the set of global solutions for $p>p_S$.