Seiichiro Wakabayashi

THE LAX-MIZOHATA THEOREM FOR NONLINEAR CAUCHY PROBLEMS*

Dedicated to Professor Kunihiko Kajitani on his sixtieth birthday.

Generalized Flows and Their Applications

We define generalized flows in order to make local results global. We study the existence domain of solutions to the Cauchy problem in the complex domain and the wave front sets of solutions to the hyperbolic Cauchy problem, using generalized flows.

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.

Generalized Hamilton Flows and Singularities of Solutions of the Hyperbolic Cauchy Problem

Publisher Summary Singularities of solutions of the hyperbolic Cauchy problem have been investigated. In these works, Hamilton flows played a key role. However, in general, Hamilton flows cannot be defined meaningfully unless the characteristic roots are smooth. To study singularities of solutions, Hamilton flows must be generalized. Results on singularities of solutions can be obtained from results on well-posedness of the Cauchy problem. To make it possible, Hamilton flows should be generalized suitably. This chapter presents a definition of generalized Hamilton flows and explains the wave front sets of solutions of the hyperbolic Cauchy problem in the framework of C ∞ , Gevrey classes, or the space of real analytic functions.

The Mixed Problem for Hyperbolic Systems

Lax [27] and Mizohata [34] proved that for the non-characteristic Cauchy problem to be C∞ well-posed it is necessary that the characteristic roots are real. In the mixed problem Kajitani [23] obtained the results corresponding to those in the Cauchy problem under some restrictive assumptions. In §3 we shall relax his assumptions (see, also, [52]). We note that well-posedness of the mixed problems has been investigated by many authors ([1], [2], [19], [20], [22], [26], [32], [33], [38]). We shall consider the mixed problem for hyperbolic systems with constant coefficients in a quarter-space in §§4–6. Hersh [13], [l4], [15] studied the mixed problem for hyperbolic systems with constant coefficients. He gave the necessary and sufficient condition for the mixed problem to be C∞ well-posed. However, his proof seems to be incomplete (see [25]). Sakamoto [37] justified his results for single higher order hyperbolic equations (see, also, [40], [41] [42]). In §4 we shall consider C∞ well-posedness of the mixed problem. Duff [11] studied the location and structures of singularities of the fundamental solutions of the mixed problems for single higher order hyperbolic equations, using the stationary phase method.

Remarks on hyperbolic systems of first order with constant coefficient characteristic polynomials

In this paper we shall deal with hyperbolic systems of first or der with constant coefficient characteristic polynomials and give a necessar y and sufficient condition for the Cauchy problem to be C1 well-posed under the maximal rank condition (see the condition (R) below).

On Hypoellipticity of the Operator\exp \left[ { - {{\left| {{x_1}} \right|}^{ - \sigma }}} \right]D_1^2 + x_1^4D_2^2 + 1

Let \( L\left( {x,D} \right) = f\sigma \left( {{x_1}} \right)D_1^2 + x_1^4D_2^2 + 1,\) Where \(x = \left( {{x_1},{x_2}} \right) \in {\mathbb{R}^2},\sigma >0\) And \( {f_\sigma }\left( t \right) = \exp \left[ { - {{\left| t \right|}^{ - \sigma }}} \right]\) and \( {f_\sigma }\left( 0 \right) = 0.\) We shall prove that L(x,D) is hypoelliptic at x=(0,0) if and only if \(\sigma< 2.\)

Propagation of Singularities for Hyperbolic Mixed Problems

Duff [4] studied the location and structures of singularities of fundamental solutions for hyperbolic mixed problems with constant coefficients in a quarter-space making use of the method of stationary phase. Deakin [3] treated first order hyperbolic systems by the same method. However, it seems that it is difficult to apply the method to the study of fundamental solutions for more general hyperbolic mixed problems. Matsumura [7] gave an inner estimate of the location of singularities of fundamental solutions which correspond to main reflected waves, making use of the localization method developed by Atiyah, Bott and Garding [1] and Hormander [5]. A localization theorem describing the location of singularities of fundamental solutions which correspond to lateral waves was obtained by the author [13] under some restrictive assumptions. In [14] the author proved a localization theorem describing the location of singularities of fundamental solutions which correspond to main reflected waves, lateral waves and boundary waves. Tsuji [12] also studied the same problem in the cases where operators are homogeneous and obtained similar results. On the other hand outer estimates of the location of singularities of fundamental solutions were given in [15] by the same method as in [1] which treated the Cauchy problems.