The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.
MSC: 35G10 35B65 35G25 35E05 Keywords: Damped beam equation Cauchy problem Fourth order wave equation with damping Asymptotic profile PerturbationWe study the initial value problem for some semilinear damped beam equation. In Takeda and Yoshikawa (submitted for publication) [9] unique global existence of a decaying solution for the problem and the smoothing effect of the solution was shown. In this article we shall give the asymptotic profiles of the solution. As a result, we observe that the decay estimates are optimal. Moreover, considering the higher order expansion of the solution, we observe more detailed information such as the contribution of the nonlinear term to the solution as t → ∞.
SUMMARYIn this article a stability result for the Falk model system is proven. The Falk model system describes the martensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal elliptic problem. We show the dynamical stability for the linearized stable critical point of the corresponding functional.
SUMMARYWe show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non-linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23:299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation.
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