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

### A. Vidal-López

Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China### R. Laister

Department of Engineering Design and Mathematics, University of the West of England, Bristol BS16 1QY, UK### J.C. Robinson

Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK### M. Sierżęga

Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK

## Abstract

We consider the scalar semilinear heat equation $u_{t}−Δu=f(u)$, where $f:[0,∞)→[0,∞)$ 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 $L_{q}(Ω)$ for all non-negative initial data $u_{0}∈L_{q}(Ω)$, when $Ω⊂R_{d}$ is a bounded domain with Dirichlet boundary conditions. For $q∈(1,∞)$ this holds if and only if $limsup_{s→∞}s_{−(1+2q/d)}f(s)<∞$; and for $q=1$ if and only if $∫_{1}s_{−(1+2/d)}F(s)ds<∞$, where $F(s)=sup_{1≤t≤s}f(t)/t$. This shows for the first time that the model nonlinearity $f(u)=u_{1+2q/d}$ is truly the ‘boundary case’ when $q∈(1,∞)$, but that this is not true for $q=1$.

The same characterisations hold for the equation posed on the whole space $R_{d}$ provided that $limsup_{s→0}f(s)/s<∞$.

## Cite this article

A. Vidal-López, R. Laister, J.C. Robinson, M. Sierżęga, A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 6, pp. 1519–1538

DOI 10.1016/J.ANIHPC.2015.06.005