Singular Phenomena of Solutions for Nonlinear Diffusion Equations involving $p(x)$-\hbox{Laplacian} Operator

TL;DR

This paper investigates the singular behaviors such as finite-time vanishing and blow-up of solutions to nonlinear diffusion equations with a variable exponent p(x)-Laplacian, highlighting the influence of p(x) and initial energy.

Contribution

It introduces the critical exponents for extinction and blow-up phenomena in solutions involving the p(x)-Laplacian, advancing understanding of variable exponent nonlinear diffusion equations.

Findings

Identified conditions for finite-time vanishing and blow-up.

Derived critical exponents for solution behaviors.

Analyzed the impact of p(x) and initial energy on solutions.

Abstract

The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $\hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-\hbox{Laplacian} operator and a nonlinear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.