Decay Estimates for Solutions of Porous Medium Equations with Advection

TL;DR

This paper establishes decay estimates for bounded weak solutions of a class of degenerate parabolic equations with advection, showing they decrease over time under broad conditions and deriving explicit decay rates.

Contribution

It provides new decay estimates and rates for solutions of porous medium equations with advection, extending previous results to more general flux conditions.

Findings

Solutions decay to zero over time.

Decay rates depend on the advection flux conditions.

Results apply to broad classes of degenerate parabolic equations.

Abstract

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form \begin{equation} \notag u_t \,+\; \mbox{div}\,f(x,t,u) \;=\; \mbox{div}\,(\;\!|\,u\,|^{\alpha} \, \nabla u \;\!), \quad \;\; x \in \mathbb{R}^{n}\!\:\!, \; t > 0, \end{equation} where $ \alpha > 0 \, $ is constant, decrease to zero, under fairly broad conditions on the advection flux $f$. Besides that, we derive a time decay rate for these solutions.