Critical Fujita exponent for a semilinear heat equation with degenerate coefficients

TL;DR

This paper establishes the existence of a critical Fujita exponent for a class of semilinear heat equations with degenerate coefficients, extending the theory to weighted cases with singularities and connections to fractional Laplacians.

Contribution

It introduces a comprehensive analysis of the Fujita exponent for heat equations with degenerate weights, including singularities and fractional Laplacian relations.

Findings

Existence of a critical Fujita exponent for weighted semilinear heat equations.

Extension of the theory to weights with singularities at the origin or along lines.

Connection between degenerate coefficients and fractional Laplacian problems.

Abstract

We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.