Porous medium equation and fast diffusion equation as gradient systems. (English) Zbl 1363.35083

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.


35G25 Initial value problems for nonlinear higher-order PDEs
47J35 Nonlinear evolution equations
47H99 Nonlinear operators and their properties
34G20 Nonlinear differential equations in abstract spaces
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[1] V. Barbu: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics, Berlin, Springer, 2010. · Zbl 1197.35002
[2] S. Boussandel: Global existence and maximal regularity of solutions of gradient systems. J. Differ. Equations 250 (2011), 929–948. · Zbl 1209.47020 · doi:10.1016/j.jde.2010.09.009
[3] H. Brézis: Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations (E. Zarantonello, ed.). Contrib. nonlin. functional Analysis. Proc. Sympos. Univ. Wisconsin, Madison, Academic Press, New York, 1971, pp. 101–156.
[4] H. Brézis: 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. (In French.)
[5] R. Chill, E. Fašangová: Gradient Systems-13th International Internet Seminar. Matfyzpress, Charles University in Prague, 2010.
[6] I. Cioranescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Mathematics and Its Applications 62, Dordrecht, Kluwer Academic Publishers, 1990. · Zbl 0712.47043
[7] V. Galaktionov, J. L. Vázquez: 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.
[8] F. Otto: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equations 26 (2001), 101–174. · Zbl 0984.35089 · doi:10.1081/PDE-100002243
[9] A. Pazy: The Lyapunov method for semigroups of nonlinear contractions in Banach spaces. J. Anal. Math. 40 (1981), 239–262. · Zbl 0507.47042 · doi:10.1007/BF02790164
[10] P. Souplet: Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations. Commun. Partial Differ. Equations 24 (1999), 951–973. · Zbl 0926.35064 · doi:10.1080/03605309908821454
[11] J. L. Vázquez: The Porous Medium Equation, Mathematical Theory. Oxford Mathematical Monographs; Oxford Science Publications, Oxford University Press, 2007.
[12] W. P. Ziemer: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120, Springer, 1989.
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