We show that the Porous Medium Equation and the Fast Diffusion Equation,u − ∆ u m = f , with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H −1 (Ω) and obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ⊆ R n and do not require any boundary regularity. Moreover the approach is used to discuss the asymptotic behaviour and order preservation of solutions.MSC 2010: 35G25, 47J35, 47H99, 34G20 Keywords: Porous medium equation, gradient system, fast diffusion, asymptotic behaviour, order preservation
IntroductionThe main object of this paper is to present a treatment of the porous medium equation and the fast diffusion equation (abbreviated PME/FDE)as a gradient system in a functional analytic framework. A gradient system is an evolution equatioṅin a Hilbert space H, with a Gelfand triple V → H → V , where V is a reflexive Banach space. The 'energy functional' E : V → R and the 'forcing term' f (a function with values in H) are given, and u (with values in V ) is the solution. We refer to Section 1 for details, and we refer to [5] for the theory of gradient systems.Let Ω ⊆ R n be open and bounded, and let H := H −1 (Ω) be the dual of the Sobolev space(Ω) densely, and with E : V → R defined bywe show that the setting of gradient systems can be used for the PME/FDE, in order to obtain solutions u : [0, ∞) → V of Cauchy problems for (0.1), with initial values u 0 ∈ V and forcing terms f : [0, ∞) → H. It is implicit in the setup that the solution has the property that u(t) m ∈ H 1 0 (Ω) a.e., and in this sense satisfies (generalised) Dirichlet boundary values zero.We note that this setting yields a unified treatment for the PME and the FDE (including the heat equation). The existence, uniqueness and asymptotics of solutions of the PME/FDE will be obtained by purely functional analytic arguments; no arguments using elliptic regularity theory, comparison principles or smoothness of the boundary of Ω are needed.We treat the PME/FDE without any restriction on the parameter m and allow all bounded, but also unbounded Ω, provided a Poincaré inequality holds. Our results on the asymtotic decay rely on the requirement that L m+1 (Ω) should be continuously embedded into H −1 (Ω). This holds for all Ω with finite measure, provided m (n − 2)/(n + 2), and for unbounded Ω in certain cases of fast diffusion ((n − 2)/(n + 2) m 1). The importance of the space H −1 (Ω) for the treatment of evolution equations of porous medium type(with a suitable 'maximal monotone graph' β) was recognized in [3]. Solutions of initial value problems in this context are obtained using the theory of evolution equations involving accretive operators; see for example [3; Thm. 21], and more recently [1; Thm. 53]. The perception that the porous medium equation (0.2) can be treated in the context of gradient systems gives an alternative functional analytic access to the solution theory. Moreover the additional structure obtained by the energy functional E yields possibiliti...
