Porous medium equation and fast diffusion equation as gradient systems

We show that the Porous Medium Equation and the Fast Diffusion Equation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot u - \Delta {u^m} = f$$\end{document}, with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H−1(Ω), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ⊆ ℝn and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.