On the Cauchy problem for a class of hyperbolic operators whose coefficients depend only on the time variable
The main goal of the paper is the study of \(C^\infty\) well-posedness for Cauchy problems for linear hyperbolic partial differential equations with time-dependent coefficients. The coefficients of the pricipal part are supposed to be real analytic. The hyperbolic partial differential equation is not necessarily strictly hyperbolic. Consequently, multiple characteristics are allowed. To prove well-posedness so-called Levi conditions are required. The author studies two scenarios. On the one hand he allows double characteristics only for a given linear hyperbolic operator, on the other hand he treats the general case of a hyperbolic operator of third order. In both cases the author proposes Levi conditions by using sub-principal symbol (general case with double characteristics) and sub-principal and sub-sub-principal symbols (special case of third order). To prove \(C^\infty\) well-posedness the author defines a family of ``approximating strictly hyperbolic Cauchy problems'' and derives uniform energy estimates of its solutions.
scientific article; zbMATH DE number 6486345