The Cauchy problem for doubly degenerate parabolic equations with weights

TL;DR

This paper studies a class of doubly degenerate parabolic equations with space-dependent weights, establishing explicit decay, support, and propagation estimates for solutions under certain conditions.

Contribution

It provides new explicit estimates for solutions of weighted doubly degenerate parabolic equations, including decay rates and finite speed of propagation.

Findings

Solutions exhibit explicit decay rates at infinity.

Finite speed of propagation is established.

Support estimates are derived explicitly in terms of the weight.

Abstract

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We assume here the solutions of the Cauchy problem to be globally integrable in space (in the appropriate weighted sense) and non-negative. Under suitable assumptions, we prove for the solutions sup estimates, i.e., the decay rate at infinity, the property of finite speed of propagation and support estimates. All our estimates are given explicitly in terms of the weight appearing in the equation.