Kimitoshi Tsutaya

Normal form and global solutions for the Klein-Gordon-Zakharov equations

Abstract In this paper we study the global existence and asymptotic behavior of solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in three space dimensions. We prove that for small initial data, there exist the unique global solutions of the Klein-Gordon-Zakharov equations. We also show that these solutions approach asymptotically the free solutions as t → ∞. Our proof is based on the method of normal forms introduced by Shatah [12], which transforms the original system with quadratic nonlinearity into a new system with cubic nonlinearity.

Nonlinear Schrödinger Equation with Inhomogeneous Dirichlet Boundary Data

In this article we study the following nonlinear Schrodinger equation iut=Δu−g∣u∣p−1u in a domain Ω⊂Rn with initial condition u(x,0)=ϕ(x) and the Dirichlet boundary condition u(x,t)=Q(x,t) on ∂Ω, where ϕ, Q are given smooth functions. The nonlinear term contributes a negative term to the energy (i.e., g<0). We present the existence theorem for a global solution of finite energy when p⩽1+2∕n.

Scattering theory for the Dirac equation with a non-local term

Consider a scattering problem for the Dirac equation with a non-local term including the Hartree type, say the cubic convolution term. We show the existence of scattering operators for small initial data in the subcritical and critical Sobolev spaces.

The generalized Ginzburg-Landau equation: posed in a quarter plane

This paper is concerned with a generalized 1D Ginzburg-Landau equation involving a fifth order term and two nonlinear terms containing spatial derivatives. The equation is posed in a quarter plane 0=