In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound that decreases with respect to time and coincides original at $t = 0$ as $t$ tends $\infty$. This can also be viewed nonlocal-in-time modified energy, summation an accumulation term due memory effect fractional derivative. particular, indicates indeed decays w.r....

The Cahn–Hilliard and viscous Cahn–Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

Abstract We develop a constrained theory for constituent migration in bodies with microstructure described by scalar phase field. The distinguishing features of the stem from systematic treatment and characterization reactions needed to maintain internal constraint given coincidence mass fraction also boundary conditions situations which interface between body its environment is structureless c...

In this paper we study a model for phase segregation consisting in a sistem of a partial and an ordinary differential equation. By a careful definition of maximal solution to the latter equation, this system reduces to an Allen-Cahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the ω-limit set is given.

The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main part ...

We study the dynamics of bubbles in the one dimensional Cahn-Hilliard equation. For a gas of diluted bubbles we find ordinary differential equations describing their interaction which permits us to describe the ulterior dynamics of the system in very good agreement with numerical simulations.

For the Allen-Cahn equation, it is well known that there a monotone standing wave joining with balanced wells of potential. In this paper we study existence traveling solutions for equation on an infinite channel. Such possess large number oscillation and they are obtained aid variational arguments.

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or compact manifold. Moreover, we prove monotonicity formula of the associated with parabolic Allen-Cahn equations. If ambient manifold is non-negative sectional and parallel Ricci curvature, this consequence new equation. As an application, show that when initial data small enough (less than twice energy one-dimensi...