# Article

 Title: Porous medium equation and fast diffusion equation as gradient systems (English) Author: Littig, Samuel Author: Voigt, Jürgen Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 65 Issue: 4 Year: 2015 Pages: 869-889 Summary lang: English . Category: math . Summary: 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. (English) Keyword: porous medium equation Keyword: gradient system Keyword: fast diffusion Keyword: asymptotic behaviour Keyword: order preservation MSC: 34G20 MSC: 35G25 MSC: 47H99 MSC: 47J35 idZBL: Zbl 06537697 idMR: MR3441322 DOI: 10.1007/s10587-015-0214-1 . Date available: 2016-01-13T09:01:16Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/144779 . Reference: [1] Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces.Springer Monographs in Mathematics Berlin, Springer (2010). Zbl 1197.35002, MR 2582280 Reference: [2] Boussandel, S.: Global existence and maximal regularity of solutions of gradient systems.J. Differ. Equations 250 (2011), 929-948. Zbl 1209.47020, MR 2737819, 10.1016/j.jde.2010.09.009 Reference: [3] Brézis, H.: Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations.Contrib. nonlin. functional Analysis. Proc. Sympos. Univ. Wisconsin, Madison Academic Press, New York (1971), 101-156 E. Zarantonello. Zbl 0278.47033, MR 0394323 Reference: [4] Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert.North-Holland Mathematics Studies Amsterdam-London: North-Holland Publishing Comp.; New York, American Elsevier Publishing Comp. (1973), French. Zbl 0252.47055, MR 0348562 Reference: [5] Chill, R., Fašangová, E.: Gradient Systems---13th International Internet Seminar.Matfyzpress Charles University in Prague (2010). Reference: [6] Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems.Mathematics and Its Applications 62 Dordrecht, Kluwer Academic Publishers (1990). Zbl 0712.47043, MR 1079061 Reference: [7] Galaktionov, V., Vázquez, J. L.: A Stability Technique for Evolution Partial Differential Equations. A Dynamical Systems Approach.Progress in Nonlinear Differential Equations and Their Applications 56 Boston, MA: Birkhäuser (2004). Zbl 1065.35002, MR 2020328 Reference: [8] Otto, F.: The geometry of dissipative evolution equations: the porous medium equation.Commun. Partial Differ. Equations 26 (2001), 101-174. Zbl 0984.35089, MR 1842429, 10.1081/PDE-100002243 Reference: [9] Pazy, A.: The Lyapunov method for semigroups of nonlinear contractions in Banach spaces.J. Anal. Math. 40 (1981), 239-262. Zbl 0507.47042, MR 0659793, 10.1007/BF02790164 Reference: [10] Souplet, P.: Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations.Commun. Partial Differ. Equations 24 (1999), 951-973. Zbl 0926.35064, MR 1680893, 10.1080/03605309908821454 Reference: [11] Vázquez, J. L.: The Porous Medium Equation, Mathematical Theory.Oxford Mathematical Monographs; Oxford Science Publications Oxford University Press (2007). Zbl 1107.35003, MR 2286292 Reference: [12] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation.Graduate Texts in Mathematics 120 Springer (1989). Zbl 0692.46022, MR 1014685 .

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