Akihiro Shimomura

Asymptotic Behavior of Solutions for Schrödinger Equations with Dissipative Nonlinearities

The asymptotic behavior in time of small solutions for the initial value problem of the Schrödinger equation with dissipative nonlinearity is studied. The nonlinearity is −λ|u|2/n u, where λ is a complex constant such that and the space dimension n = 1, 2, or 3. This nonlinearity is critical between the short range interaction and the long range one. If , then the nonlinearity has a dissipative property. The main purpose of this article is to show that in the case of , there exists a unique global solution for this initial value problem which decays like (t log t)−n/2 as t → +∞ in L ∞ for small initial data, and to obtain the large time asymptotics of it.

SCATTERING THEORY FOR ZAKHAROV EQUATIONS IN THREE-DIMENSIONAL SPACE WITH LARGE DATA

We study the scattering theory for the Zakharov equation in three-dimensional space. We show the unique existence of the solution for this equation which tends to the given free profile with no restriction on the size of the scattered states and on the support of the Fourier transform of them. This yields the existence of the pseudo wave operators.

Scattering theory for the coupled Klein-Gordon-Schrödinger equations in two space dimensions II

We study the scattering theory for the coupled Klein- Gordon-Schrodinger equation with the Yukawa type interaction in two space dimensions.The scattering problem for this equation belongs to the borderline between the short range case and the long range one. We show the existence of the wave operators to this equation without any size restriction on the Klein-Gordon component of the final state. 2 . (KGS) Here u and v are complex and real valued unknown functions of (t, x) ∈ R × R 2 , respectively.In the present paper, we prove the existence of the wave operators to the equation (KGS) without any size restriction on the Klein-Gordon component of the final state. A large amount of work has been devoted to the asymptotic behavior of solutions for the nonlinear Schrodinger equation and for the nonlinear Klein- Gordon equation.We consider the scattering theory for systems centering on the Schrodinger equation, in particular, the Klein-Gordon-Schrodinger, the Wave-Schrodinger and the Maxwell-Schrodinger equations.In the scat- tering theory for the linear Schrodinger equation, (ordinary) wave operators