Inhomogeneous parabolic equations on unbounded metric measure spaces

TL;DR

This paper extends the analysis of inhomogeneous semilinear parabolic equations to general metric measure spaces, providing conditions for the existence and non-existence of solutions using Harnack inequalities and estimates.

Contribution

It generalizes previous Euclidean space results to metric measure spaces, offering new theoretical tools for analyzing such equations.

Findings

Established Harnack-type inequalities in time

Provided sufficient conditions for solution existence and non-existence

Generalized results from Euclidean spaces to metric measure spaces

Abstract

We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence, and global existence of weak solutions. This paper generalizes previous results on Euclidean spaces to general metric measure spaces.