Sharp analytic-Gevrey regularity estimates down to $t=0$ for solutions to semilinear heat equations

Abstract

We study the Gevrey regularity down to $t=0$ of solutions to the initial-value problem for the semilinear heat equation $\partial_tu-\Delta u+F[u]=0$ with polynomial non-linearities. The approach is based on suitable iterative fixed point methods in $L^p$-based Banach spaces with anisotropic Gevrey norms with respect to the time and space variables. We also construct explicit solutions uniformly analytic in $t\geq 0$ and $x\in {\mathbb R}^n$ for some conservative non-linear terms with symmetries.