A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

TL;DR

This paper fully characterizes the nonlinear functions for which the semilinear heat equation admits local solutions in Lebesgue spaces, revealing the precise boundary between existence and non-existence based on the growth of the nonlinearity.

Contribution

It provides a complete characterization of local existence for semilinear heat equations in Lebesgue spaces, including the critical boundary case for the model nonlinearity.

Findings

The boundary case for $f(u)=u^{1+2q/d}$ is confirmed for $q>1$.

The characterization depends on the growth conditions of $f$ at infinity.

Results extend to the whole space with additional conditions.

Abstract

We consider the scalar semilinear heat equation $u_t-\Delta u=f(u)$, where $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing but need not be convex. We completely characterise those functions $f$ for which the equation has a local solution bounded in $L^q(\Omega)$ for all non-negative initial data $u_0\in L^q(\Omega)$, when $\Omega\subset{\mathbb R}^d$ is a bounded domain with Dirichlet boundary conditions. For $q\in(1,\infty)$ this holds if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and for $q=1$ if and only if $\int_1^\infty s^{-(1+2/d)}F(s)\,{\rm d}s<\infty$, where $F(s)=\sup_{1\le t\le s}f(t)/t$. This shows for the first time that the model nonlinearity $f(u)=u^{1+2q/d}$ is truly the `boundary case' when $q\in(1,\infty)$, but that this is not true for $q=1$. The same characterisation results hold for the equation posed on the whole space ${\mathbb R}^d$ provided that in addition $\limsup_{s\to0}f(s)/s<\infty$.