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Porous medium equation and fast diffusion equation as gradient systems

Samuel Littig, Jürgen Voigt (2015)

Czechoslovak Mathematical Journal

We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u} - Δ u^{m} = f$ , with $m \in (0, \infty)$ , 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 $Ω \subseteq ℝ^{n}$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

Positive periodic solutions for the Korteweg-de Vries equation.

Georgiev, Svetlin Georgiev (2007)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon

Alain Bachelot (1988)

Annales de l'I.H.P. Physique théorique

Problème de Cauchy pour des systèmes hyperboliques semi-linéaires

Alain Bachelot (1984)

Annales de l'I.H.P. Analyse non linéaire

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

We investigate the propagation of the uniform spatial Gevrey $G^{σ}$ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.