Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity

TL;DR

This paper investigates the conditions under which solutions exist or do not exist for the Hardy parabolic equation with general nonlinearity, establishing optimal initial data integrability criteria.

Contribution

It provides the first comprehensive analysis of local existence and nonexistence for this class of equations with general nonlinearities, using the supersolution method.

Findings

Established optimal integrability conditions for initial data.

Proved local existence of nonnegative solutions under these conditions.

Identified scenarios leading to nonexistence of solutions.

Abstract

In this paper, we consider the Cauchy problem for the Hardy parabolic equation with general nonlinearity and establish the local existence and nonexistence results. Our results provide the optimal integrability conditions on initial function for the existence of a local-in-time nonnegative solution. The proof of the existence result is based on the supersolution method.