Porous medium equation and fast diffusion equation as gradient systems
Samuel Littig, Jürgen Voigt · Czechoslovak Mathematical Journal · 2015
We show that the Porous Medium Equation and the Fast Diffusion Equation, $$\dot u - \Delta {u^m} = f$$ , 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 ⊆ ℝ n and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.