Initial traces and solvability for a semilinear heat equation on a half space of ${\mathbb R}^N$

TL;DR

This paper investigates the initial traces and solvability conditions for nonnegative solutions to a semilinear heat equation on a half space, establishing existence, uniqueness, and sharp criteria for initial data.

Contribution

It provides the first comprehensive analysis of initial traces and solvability criteria for semilinear heat equations on half spaces with zero Dirichlet boundary conditions.

Findings

Existence and uniqueness of initial traces for solutions.

Necessary and sufficient conditions for solvability of the Cauchy-Dirichlet problem.

Identification of optimal singularities of initial data.

Abstract

We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.