Masayoshi Tsutsumi

On solutions of some doubly nonlinear degenerate parabolic equations with absorption

Abstract The existence, uniqueness, regularity, and behavior of solutions to the initial-boundary value problem for the equation (U β ) t − ∑ i=1 N ∂ ∂x i ∂u ∂x i p−2 ∂u ∂x i +λu γ =0 , with zero-Dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in R N, where β, γ, λ ⩾ 0 and p >1.

On coupled Klein-Gordon-Schrödinger equations, II

A classical model that describes a system of conserved scalar nucleons interacting with neutral scalar mesons is considered. The dynamics of these fields are coupled through the Yukawa interaction. In the case of relativistic fields, that is, when nucleons are described by Dirac spinor fields, the coupled Klein-Gordon-Dirac equations are encountered. Both Klein-Gordon-Dirac equations and Klein-Gordon-Schroedinger equations have energy forms indefinite in nature. The initial-boundary value problem for the coupled Klein-Gordon-Schroedinger equations in three space dimensions is discussed here. The purpose is to establish the existence and uniqueness theorems of global C/sup x/-solutions of the initial-boundary value problem. The main difficulties lie in the proof of the existence of strong solutions and their regularity properties. (RWR)

On smooth solutions to the initial-boundary value problem for the nonlinear Schro¨dinger equation in two space dimensions

On considere le probleme aux valeurs limites et initiales: iu t -Δu+f(|u| 2 )u=0, x∈Ω, t>0, u(x,0)=u 0 (x), x∈Ω, u/ ∂Ω =0 t>0, ou Ω est un domaine borne de R 2 a frontiere lisse ∂Ω

The Cauchy problem for the coupled Maxwell–Schrödinger equations

The Cauchy problem for the coupled Maxwell–Schrodinger equations in Rd in the Lorentz gauge is considered. The viscosity method is used to establish local existence. In one and two space dimensions, global solutions are obtained.

On the inverse scattering problem for the one-dimensional Schrödinger equation with an energy dependent potential

In this paper we are concerned with the scattering problem for the onedimensional Schriidinger equation

On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations

where a > 0 and Q is a bounded domain in RN with smooth boundary X?. When a = 0 (undamped case), the global existence of smooth solutions to the initial-boundary value problem (l)-(3) has been proved only when N=2 and f(lu12)w lulp with p = 2 (Brezis and Gallouet Cl]), with 2 I p I3 (M. Tsutsumi [9]). Higher dimensional cases are very difficult pribl&s and remain unproved as far as the author knows. If we add a linear damping term iau (a > 0), then the mass Ilull iZtR1 and the energy { IVu(x, t)l* dx + j F( lu(x, t)l*) dx decay exponentially as t + cc (as is seen below), which may assert the global existence of (small) smooth solutions even if the nonlinear term f(lul’)u does not have a proper sign. It is our purpose to establish the global existence and decay estimates of smooth solutions of the problem (l)-(3) with small data. There are a number of works on the nonlinear evolution equations (see [3, 5-71 and their references) in the following spirit: “Energy and decay estimates give global existence of solutions.” If the damping term is not added, there is no hope that the decay estimates of solutions in some F(Q) (p 2 2) can be obtained since

On existence of global solutions of Schrödinger equations with subcritical nonlinearity for ̂^{}-initial data

We construct a local theory of the Cauchy problem for the nonlinear Schrödinger equations iut + uxx ± |u|α−1u = 0, x ∈ R, t ∈ R, u(0, x) = u0(x) with α ∈ (1, 5) and u0 ∈ ̂ Lp(R) when p lies in an open neighborhood of 2. Moreover we prove the global existence for the initial value problem when p is sufficiently close to 2.

Scattering of Solutions of Nonlinear Klein-Gordon Equations in Higher Space Dimensions

Publisher Summary The scattering theory for nonlinear Klein–Gordon equations has been developed by many authors (such as, Segal, Strauss, Reed, and others). This chapter aims to extend recent results of Strauss on low energy scattering. In general useful methods by which one attacks nonlinear hyperbolic problem are energy estimates, L p –L q (decay) estimates for linear problem, and estimates of nonlinearity in various function spaces (e.g., Sobolev spaces, Besov spaces). The methods employed in this chapter are the same. The difficulty is the suitable choice of the spaces in which solutions of Nonlinear Klein-Gordon Equation (NLKG) lie.

The Meissner effect and the Ginzburg-Landau equations in the presence of an applied magnetic field

It is shown that a way of phenomenological description of the Meissner effect in a superconductor under an applied magnetic field is to consider the minimizing problem of the Gibbs free energy under the constraints of complete expulsion of magnetic field from the superconducting states.

A mathematical formulation of the “shicho” in the game of Go

The game of Go is a zero-sum game by two players with perfect information. The game of Go has a graph theoretical structure and tactics. BW graph model is a mathematical model for the game of Go, and represents postions and games graph theoretically. This model uses degrees of intersections on the board. In this paper, we formulate a pattern of stones, shicho. By our result, we can represent shicho states and determine these results statically.