Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian

TL;DR

This paper investigates the existence, blow-up, and stability of solutions for a non-local fractional p(x)-Laplacian diffusion equation with variable exponents, using advanced variational and potential well methods.

Contribution

It introduces new analytical techniques to establish local and global solutions, as well as finite time blow-up, for a complex non-local PDE with variable exponents.

Findings

Existence of local solutions via sub-differential approach

Global solutions and blow-up characterized using potential well and Nehari manifold

Asymptotic stability of solutions in variable exponent Lebesgue spaces

Abstract

In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.