The Cauchy problem for effectively hyperbolic equations (a special case)
This paper is the first step to prove the proposition that effectively hyperbolic operators are strongly hyperbolic. (The converse has been proved by V. Ya. Ivrij and V. Petkov.) Here it is proved by the factorization method that the Cauchy problem for a hyperbolic equation of second order is well posed if it satisfies an assumption. It will be shown in a forthcoming paper for the standard type of the problem in the title [Publ. Res. Inst. Math. Sci. 20 (1984) (to appear)], that any effectively hyperbolic operator of second order satisfies this assumption. The last part of this work (to appear) deals with the general case to complete the proposition for single equations of higher order by reduction to the previous cases and to extend it to non-linear equations by means of the Nash-Moser implicit function theorem. [Refer to RIMS Kokyuroku 468, Kyoto Univ. (1984), or to a note by the author in Proc. Symp. Pure Math. (1983) on Nonlinear Functional Analysis and Applications.]
scientific article; zbMATH DE number 3869838