Energy solutions of the Cauchy-Neumann problem for porous mediumequations

The existence of energy solutions to the Cauchy-Neumann problem for the porous medium equation of the form vt − ∆(|v|m−2v) = αv with m ≥ 2 and α ∈ R is proved, by reducing the equation to an evolution equation involving two subdifferential operators and exploiting subdifferential calculus recently developed by the author.


1.
Introduction. Let us consider the existence of solutions u = u(x, t) to the Cauchy-Neumann problem (CNP) for the porous medium equation, where α ∈ R, m ≥ 2, T > 0, Ω is a bounded domain of R N with smooth boundary ∂Ω and N ∈ N.
As for Cauchy-Dirichlet problems, two major nonlinear semigroup approaches are widely used to treat porous medium type equations such as with a maximal monotone graph β in R × R. One is an "L 1 -framework" based on the m-accretive operator theory. Brézis and Strauss [6] proved the m-accretivity in X := L 1 (Ω) of the operator A : X → X given by Au := −∆β(u) equipped with the homogeneous Dirichlet boundary condition. Hence due to an abstract theory of Crandall and Liggett, the operator A generates a continuous contraction semigroup S(t) in X, and moreover, S(t)v 0 is the unique generalized solution of the Cauchy-Dirichlet problem for (1). The other is an ''H −1 -framework" based on the subdifferential operator theory. Let X := H −1 (Ω) be a Hilbert space with the inner product (u, v) H −1 (Ω) := ∂u ∂n and their results enable us to treat (1) with (2) for any maximal monotone graph β (see also [11]). The (H 1 ) * -framework is also applied to the Neumann boundary condition by Damlamian [7] (in case β is Lipschitz continuous) and by Kubo-Lu [10] (in case β(v) = −1/v), however, some restrictions are always imposed on β, and there seems to be no contribution of subdifferential approach which can cover the Cauchy-Neumann problem for the porous medium equation (i.e., β(u) = |u| m−2 u). Framework Base space X Boundary condition Nonlinearity of β L 1 L 1 (Ω) D, N, R any for "D" In studies of the asymptotic behavior of solutions for (CNP), energy identities (or inequalities) play crucial roles. However, the semigroup approaches described above have not been designed so well that one can directly obtain energy inequalities sufficient for the analysis.
The purpose of this paper is to prove the existence of weak solutions for (CNP) and derive energy inequalities even for α = 0 by performing subdifferential calculus. We particularly exploit techniques recently developed by the author [1,2] for nonlinear evolution equations involving two subdifferential operators in reflexive Banach spaces. and put H = L 2 (Ω) with the norm | · | H and the inner product (·, ·) H . Then

Reduction to an evolution equation. Set
Let us introduce the following Cauchy problem for an evolution equation as a weak form of (CNP), dv Then ∂ V ϕ(u) and ∂ H ψ(u) coincide with Au and |u| m −2 u(·) respectively (here and henceforth, we write |r| q r = |r| q+1 sgn(r) for r ∈ R and q > −1). Therefore by We shall treat (5)- (7) to discuss the existence of solutions for (CNP).
3. Main result. Define a function J : We denote by C w ([0, T ]; H 1 (Ω)) the set of all H 1 (Ω)-valued weakly continuous functions on [0, T ]. We are concerned with solutions of (CNP) given as follows.
Our main result reads, We next prepare the following proposition.

Uniqueness. Let (u, v) and (û,v) be weak solutions of (CNP) on [0, T ] with an initial data
Since v(t), Here we used the fact that Moreover, we notice that Hence integrating both sides over (0, t) and applying Gronwall's inequality, we have Thus every weak solution of (CNP) is unique.