Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition

TL;DR

This paper investigates the initial boundary behavior of solutions to a semilinear heat equation with Dirichlet conditions, establishing conditions for existence and characterizing singularities of initial data.

Contribution

It provides sharp criteria for the existence of solutions and identifies optimal singularities for nonnegative initial data in the Dirichlet problem.

Findings

Necessary and sufficient conditions for solution existence

Characterization of optimal singularities

Identification of initial trace properties

Abstract

We study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, we obtain sharp necessary conditions and sufficient conditions on the existence of nonnegative solutions and identify optimal singularities of solvable nonnegative initial data.