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

Abstract

We consider the scalar semilinear heat equation $u_t-\mathrm{\Delta }u=f(u)$, where $f:[0,\infty )\to [0,\infty )$ 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 $L^q(\mathrm{\Omega })$ for all non-negative initial data $u_0\in L^q(\mathrm{\Omega })$, when $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a bounded domain with Dirichlet boundary conditions. For $q\in (1,\infty )$ this holds if and only if $\lim {\sup }_{s\to \infty }{s}^{-(1+2q/d)}f(s)<\infty$; and for $q=1$ if and only if ${\int }_1^{\infty }{s}^{-(1+2/d)}F(s)\mathrm{d}s<\infty$, 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 characterisations hold for the equation posed on the whole space ${\mathbb{R}}^d$ provided that $\lim {\sup }_{s\to 0}f(s)/s<\infty$.