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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.
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

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