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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: The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples. [Copyright &y& Elsevier]

Solving approximate energy flow partial differential equations for structural elements can be done analytically (energy flow analysis—EFA) or using finite element approximations (energy finite element method—EFEM). In this paper, energy equations are solved using the spectral element method (SEM). This new approach, which is called the energy spectral element method (ESEM), can be applied to predict the distribution of the energy flow and energy density of built-up structures at high frequencies. Energy spectral element method is a matrix formulation based on the general solution of the partial differential equations for energy density in structural vibrations such as longitudinal and transversal vibrations of frames. A spectral energy element can be shown to be equivalent to an infinite number of energy finite elements. In this work, numerical models involving coupled rods and beams are generated by energy spectral element method, and the results obtained are compared with energy densities computed from the displacement fields predicted by the SEM to solve the conventional boundary value problem. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the initial value problem of the OST equation (, ), where and denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space when , and globally well-posed in when . [Copyright &y& Elsevier]

Abstract: This paper proposes a unified analytical solution of the wave equation governing the propagation of the longitudinal stress wave in an elastic rod. Within a single formula derived by using the Laplace transform and inverse transform, the solution covers the contributions of the external excitations, the nonzero initial conditions and the inhomogeneous boundary conditions altogether, including such boundary conditions that the dependent variables at the ends of the rod are restricted with an equation. The proposed formula, particularly suitable for transient problems, could be regarded as an exact interpolation function in the time domain, provided the rod works as a component in a complex system, etc. Four examples are presented to show the applications of the solution. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the local side of the bifurcation from infinity at the first eigenvalue of several elliptic operators. We underline that the key to decide the local behavior of the bifurcation lies in the sign of certain integral involving all the values of the nonlinearity. [Copyright &y& Elsevier]

Abstract: In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55–81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of . This paper extends this study to the resolution of a mixed exterior Laplace''s problem. Here, we give existence, unicity and regularity results in ''s theory with , in weighted Sobolev spaces. [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: In this paper we show that, under suitable assumptions, the solutions to the approximating initial and boundary value problems (cf. introduction) converge in , for suitable indexes and , to the solution to the limit problem . The last two Sections 4–5 are devoted to a similar approximation result, in a Banach-space framework, and involve a generalization of the kernels k, h and operator A. [Copyright &y& Elsevier]

Abstract: In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition , . So the Esquivel-Avila''s results are generalized and improved. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of positive solutions for the boundary value problem of one-dimensional p-Laplacian with delay. The proof is based on the Guo–Krasnoselskii fixed-point theorem in cones. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of Lyapunov-type inequalities () for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in . It is proved that the relation between the quantities p and plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear 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]