Kunihiko Kajitani

Analytically Smoothing Effect for Schrödinger Type Equations with Variable Coefficients

We shall investigate analytically smoothing effects of the solutions to the Cauchy problem for Schrodinger type equations. We shall prove that if the initial data decay exponentially then the solutions become analytic with respect to the space variables. Let T > 0.

The Cauchy Problem for Uniformly Diagonalizable Hyperbolic Systems in Gevrey Classes

Publisher Summary This chapter discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems of linear partial differential equations in Gevrey classes. It also discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems whose coefficients are in Gevrey class. It has been proven that if the coefficients of the system are constant, the uniformly diagonalizable hyperbolic system is equivalent to strongly hyperbolic one, that is, this system is stable under the perturbation of lower order term of the system. In general, the characterization of the strong hyperbolicity in the C ∞ -sense for the systems of variable coefficients is an open problem. The uniform diagonalizability for the hyperbolic systems is a sufficient condition for the Cauchy problem to be well-posed.

Time Global Solutions to the Cauchy Problem for Multidimensional Kirchhoff Equations

The aim of this work is to get the time global solutions to the Cauchy problem in Sobolev spaces for multidimensional Kirchhoff equations.

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

AbstractIn this paper we consider the Cauchy problem for the equation

Strong Gevrey Solvability for a System of Linear Partial Differential Equations

\partial ^2 u\left( {t,x} \right)/\partial t^2 = - \sum\nolimits_{j,k = 1}^n {D_j } \left( {a_{jk} \left( {t,x} \right)D_k u\left( {t,x} \right)} \right) + f\left( {t,x} \right)

The Hyperbolic Cauchy Problem

, where the matrix {ajk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs

The Cauchy problem for Schrödinger type equations with variable coefficients

We investigate the sufficient conditions on the initial data in order that the solutions of the Cauchy problem for Schrodinger equations with some potentials and also for even dimentional wave equations decay exponentially in time.

Cauchy Problem for Non-Strictly Hyperbolic Systems

We consider a class of linear systems whose principal symbol satisfies a certain condition of semi-hyperbolicity, and we prove the local surjectivity in suitable Gevrey spaces.