A supersolution approach to doubly degenerate parabolic equations with weights

TL;DR

This paper develops a supersolution method for analyzing doubly degenerate parabolic equations with space-dependent weights, providing sharp decay bounds and extending to inhomogeneous densities through variable transformations.

Contribution

It introduces a novel supersolution approach for weighted degenerate parabolic equations, enabling precise decay estimates and applicability to inhomogeneous density cases.

Findings

Constructed supersolutions and subsolutions for weighted degenerate equations

Established sharp temporal decay bounds for solutions

Extended methods to equations with inhomogeneous densities

Abstract

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, where the exponent satisfies the doubling condition. In particular, both the so called logconvex and logconcave cases may be considered. Under the additional natural assumptions we construct supersolutions and subsolutions allowing us to control the precise sharp temporal decay bounds. We apply our results also to an equation with inhomogeneous density, via a suitable variable transformation.