We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

In this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

The Douglas-Rachford and Peaceman-Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments.

We consider some discrete and continuous dynamics in a Banach spaceinvolving a non expansive operator J and a corresponding family ofstrictly contracting operators Φ (λ, x): = λJ( $\frac{1-\lambda}{\lambda}$ x) for λ ∈ ] 0,1] . Our motivationcomes from the study of two-player zero-sum repeated games, wherethe value of the n-stage game (resp. the value of theλ-discounted game) satisfies the relationv n = Φ( $\frac{1}{n}$ , $v_{n-1}$ ) (resp.  $v_\lambda$ = Φ(λ, $v_\lambda$ )) where J is the Shapleyoperator of the game. We study the evolution equationu'(t) = J(u(t))- u(t) as well as associated Eulerian schemes,establishing a new exponential formula and a Kobayashi-likeinequality for such trajectories. We prove that the solution of thenon-autonomous evolution equationu'(t) = Φ(λ(t), u(t))- u(t) has the same asymptoticbehavior (even when it diverges) as the sequence v n (resp. as thefamily $v_\lambda$ ) when λ(t) = 1/t (resp. whenλ(t) converges slowly enough to 0).

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples. 2000 Mathematics subject classification: primary 35K.

In this paper we test the feasibility of coupling two heterogeneous mathematical modeling integrated within two different codes residing on distant sites. A prototype is developed using Schwarz type domain decomposition as the mathematical tool for coupling. The computing technology for coupling uses a CORBA environment to implement a distributed client-server programming model. Domain decomposition methods are well suited to reducing complex physical phenomena into a sequence of parallel subproblems in time and space. The whole process is easily tuned to underlying hardware requirements.