Critical exponent for the global existence of solutions to a nonlinear degenerate/singular parabolic equation

TL;DR

This paper studies a nonlinear heat equation with degenerate or singular coefficients, establishing a critical exponent that determines whether solutions exist globally or blow up, extending parabolic theory to complex coefficient scenarios.

Contribution

It introduces the existence of a Fujita exponent for equations with $A_2$ class coefficients, including singular and degenerate cases, and links certain singularities to fractional Laplacian theory.

Findings

Existence of a Fujita critical exponent for the equation.

Dichotomy between global existence and finite-time blow-up.

Connection to fractional Laplacian via Caffarelli-Silvestre extension.

Abstract

We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.