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This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation subject to the following dynamic boundary conditions: and the initial condition where Ω is a bounded domain in Rn (n⩽3) with smooth boundary Γ , and Γs>0, σs>0, gs>0, hs are given constants; Δ|| is the tangential Laplacian operator, and ν is the outward normal direction to the boundary.This problem has been considered in the recent paper by Racke and Zheng (Adv. Differential Equations 8 (1) (2003) 83) where the global existence and uniqueness were proved. In a very recent manuscript by Prüss, Racke and Zheng (Konstanzer Schrift. Math. Inform. 189 (2003)) the results on existence of global attractor and maximal regularity of solution have been obtained. In this paper, convergence of solution of this problem to an equilibrium, as time goes to infinity, is proved. [Copyright &y& Elsevier]

Abstract: This paper presents a partial classification for type-changing symplectic Monge–Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with Dirichlet boundary conditions on varying domains. We assume that the limit domain be stable in the sense of Keldyš [Amer. Math. Soc. Transl. 51 (1966) 1–73]. We further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain, and only require -convergence outside. As a consequence, uniform and -convergence are the same in the trivial case of homogenisation of a perforated domain. We are also able to deal with certain cracking domains. [Copyright &y& Elsevier]

Abstract: It is known that any periodic orbit of a Lipschitz ordinary differential equation must have period at least , where is the Lipschitz constant of . In this paper, we prove a similar result for the semilinear evolution equation : for each with there exists a constant such that if is the Lipschitz constant of as a map from into then any periodic orbit has period at least . As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier–Stokes equations with periodic boundary conditions. [Copyright &y& Elsevier]