Initial traces and solvability of porous medium equation with power nonlinearity

TL;DR

This paper investigates the initial trace properties and solvability conditions of solutions to the porous medium equation with power nonlinearity, providing necessary and sufficient criteria for existence based on initial data singularities.

Contribution

It introduces new criteria for initial trace characterization and establishes sharp conditions for the existence of solutions using advanced function spaces.

Findings

Necessary conditions for solution existence are identified.

Sharp sufficient conditions are established using Morrey spaces.

Optimal initial data singularities for solvability are characterized.

Abstract

In this paper we study qualitative properties of initial traces of solutions to the porous medium equation with power nonlinearity, and obtain necessary conditions for the existence of solutions to the corresponding Cauchy problem. Furthermore, we establish sharp sufficient conditions for the existence of solutions to the Cauchy problem using uniformly local Morrey spaces and their variations, and identify the optimal singularities of the initial data for the solvability of the Cauchy problem.