Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

TL;DR

This paper investigates the behavior of solutions to a degenerate parabolic equation with gradient-dependent absorption, establishing gradient estimates, long-term mass limits, support expansion rates, and dead core persistence.

Contribution

It introduces new gradient estimates for such equations and analyzes the interplay between diffusion and absorption, revealing long-term solution properties.

Findings

Identification of the limit of the mass as time approaches infinity

Determination of the support expansion rate for solutions

Demonstration of dead core persistence in solutions

Abstract

Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.