Decay Estimates for Solutions of Porous Medium Equations with Advection

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form ut+divf(x,t,u)=div(|u|α∇u),x∈Rn,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{t} + \operatorname{div}f(x,t,u) = \operatorname{div}\bigl( |u|^{\alpha } \nabla u\bigr), \quad x \in \mathbb{R}^{n} , \ t > 0, $$\end{document} where α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha > 0 $\end{document} is constant, decrease to zero, under fairly broad conditions on the advection flux f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f$\end{document}. Besides that, we derive a time decay rate for these solutions.