Geodesic
Porous medium equation and fast diffusion equation as gradient systems
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 869-889.

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We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot u-\Delta u^m=f$, with $m\in (0,\infty )$, can be modeled as a gradient system in the Hilbert space $H^{-1}(\Omega )$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $\Omega \subseteq \mathbb R^n$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.
DOI : 10.1007/s10587-015-0214-1
Classification : 34G20, 35G25, 47H99, 47J35
Keywords: porous medium equation; gradient system; fast diffusion; asymptotic behaviour; order preservation 
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Littig, Samuel; Voigt, Jürgen. Porous medium equation and fast diffusion equation as gradient systems. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 869-889. doi : 10.1007/s10587-015-0214-1. http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0214-1/

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