Abstract. We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain Ω ⊂ R N and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the whole of R N \ Ω). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.

This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L 2 -framework by introducing a peculiar discretization of the unidirectional diffusion equation by means of variational inequities of obstacle type and by developing a regularity theory for such variational inequalities. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions.

The asymptotic behavior of viscosity solutions to the Cauchy-Dirichlet problem for the degenerate parabolic equation u t = ∆ ∞ u in Ω × (0, ∞), where ∆ ∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) Ω = R N and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of ∆ ∞ φ = 0.

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an ε-dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as ε → 0. The argument relies on the suitable dualization of the former analysis of [G. Akagi and U. Stefanelli, J. Funct. Anal., 260 (2011), pp. 2541-2578] and results in a considerable extension of the possible application range of the WED functional approach to nonlinear diffusion phenomena, including the Stefan problem and the porous media equation.

The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for fast diffusion equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Łojasiewicz-Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Łojasiewicz-Simon inequality in a different way.

Every solution u = u(x, t) of the Cauchy-Dirichlet problem for the fast diffusion equation, ∂ t (|u| m−2 u) = ∆u in Ω × (0, ∞) with a smooth bounded domain Ω of R N and 2 < m < 2 * := 2N/(N − 2) + , vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysis of the profiles. Our method of proof relies on a detailed analysis of a dynamical system on some surface in the usual energy space as well as energy method and variational method.

The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation:∂ϕ is the so-called subdifferential operator from a real Banach space V into its dual V * . The study in the Hilbert space setting (V = V * = H : Hilbert space) is already developed in detail so far. However, the study here is done in the V -V * setting which is not yet fully pursued. Our method of proof relies on approximation arguments in a Hilbert space H . To assure this procedure, it is assumed that the embeddings V ⊂ H ⊂ V * are both dense and continuous.