Critical behavior for a semilinear parabolic equation with forcing term depending of time and space

TL;DR

This paper studies the long-term behavior of solutions to a time- and space-dependent inhomogeneous semilinear heat equation, identifying a critical exponent that determines solution existence and showing its discontinuity when the forcing term loses time dependence.

Contribution

It introduces the critical exponent for a semilinear heat equation with inhomogeneous forcing depending on time and space, revealing its discontinuity when the forcing becomes time-independent.

Findings

Identified the critical exponent separating existence and nonexistence of solutions.

Proved the discontinuity of the critical exponent at the time-independent forcing case.

Analyzed the large-time behavior of sign-changing solutions.

Abstract

We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the nonexistence/existence of global-in-time solutions, and show the discontinuity of this critical exponent at the point which the inhomogeneous term becomes independent of time variable.