Abstract
We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), with m ∈ (0, ∞), 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 ⊆ ℝn and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.
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The first named author was supported by the DFG project “Variational problems related to the 1-Laplace operator”.
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Littig, S., Voigt, J. Porous medium equation and fast diffusion equation as gradient systems. Czech Math J 65, 869–889 (2015). https://doi.org/10.1007/s10587-015-0214-1
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DOI: https://doi.org/10.1007/s10587-015-0214-1