Critical exponents for the $p$-Laplace heat equations with combined nonlinearities

TL;DR

This paper investigates the large-time behavior of solutions to a generalized p-Laplace heat equation with combined nonlinearities, revealing discontinuities in critical exponents and the influence of forcing terms on critical behavior.

Contribution

It extends previous work by analyzing the discontinuity of critical exponents and introduces a new critical exponent influenced by forcing terms.

Findings

Discontinuity of critical exponents identified.

Fills the gap in previous critical case results.

Defines a new critical exponent depending on forcing terms.

Abstract

This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.