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

joint work with Robert Laister (University of the West of England), Mikolaj Sierzega (Warwick), and Alejandro Vidal-Lopez (Xi’an Jiaotong-Liverpool University)

We consider the nonlinear heat equation $u_t-\Delta u=f(u)$ with $u(0)=u_0$, with Dirichlet boundary conditions on a bounded domain $\Omega\subset{\mathbb R}d$. We assume that $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing. We give a characterisation (an “if and only if” result) of those $f$ for which the equation has a local solution bounded in $Lq(\Omega)$ for all initial data in $Lq(\Omega)$ for all $q\in[1,\infty)$.

Versions of our proofs are also valid for the case $\Omega=\Rd$.

This talk is part of the Mathematics Colloquium series.