The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularizat...

We survey recent wellposedness and illposedness results for the Cauchy problem for nonlinear Schrödinger equations in inhomogeneous media such as Riemannian manifolds or domains of the Euclidean space, trying to emphasize the influence of the geometry. The main tools are multilinear Strichartz estimates for the Schrödinger group. Mathematics Subject Classification (2000). Primary 35Q55; Seconda...

In this paper we mainly study the Cauchy problem for the derivative nonlinear Schrödinger equation in d-dimension (d ≥ 2). We obtain some global well-posedness results with small initial data. The crucial ingredients are L e , L ∞,2 e type estimates, and inhomogeneous local smoothing estimate (L e estimate). As a by-product, the scattering results with small initial data are also obtained.

This paper presents several characterizations of a local α-times integrated C-semigroup {T(t); 0 ≤ t < τ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem.We also discuss properties concerning the nondegeneracy of T(·), the injectivity of C, the closability of subgenerators, the commutativity of T(·), and extension of solutions of the as...