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Abstract: This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system where are T-periodic functions, , . Under certain conditions, the system admits two stable semi-trivial periodic solutions and and a unique coexistence periodic solution , which is unstable and satisfies and for . In this paper we prove that the system admits a time periodic traveling wave solution connecting two periodic solutions and as , where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution is asymptotically stable and unique modulo translation for front-like initial values. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the initial value problem for the nonlinear Klein–Gordon–Schrödinger (KGS) equations in time–space. By using viscous approach, the existence of the global finite-energy solution is established for the nonlinear KGS equations by compactness argument. In addition, the uniqueness of the solution is proved by introducing a function with integral form. [Copyright &y& Elsevier]

Abstract: In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the -energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in -sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data. [Copyright &y& Elsevier]