Quasilinear hyperbolic operators with the characteristics of variable multiplicity (Q1818020)

The authors are interested in the consideration of the Cauchy problem for quasilinear weakly hyperbolic equations. They discuss the local existence of Sobolev solutions, the question for uniqueness and the stable global solvability in the case of time degeneration. They prescribe Levi conditions of \(C^\infty\)-type which are near to optimal ones. As usually the expected loss of regularity produces essential difficulties. The authors overcome these difficulties by application of a reduction scheme which reduces the starting problem to an auxiliary problem which allows to work in spaces with a special asymptotic with respect to \(t\). Corresponding linear equations are reduced to a first order pseudodifferential system, which is studied by the aid of symbols with finite smoothness. This is a consequence of the goal of the authors and the nonlinearities. Finally, the existence of a symmetrizer yields energy estimates of strictly hyperbolic type for the solutions of the auxiliary Cauchy problem. In connection with the uniqueness and stable global solvability some interesting open problems and relations to other topics of p.d.e. are formulated. The concept of this paper was developed in \textit{M. Dreher} [Local solutions to quasilinear weakly hyperbolic differential equations, PhD-thesis, Freiberg (1999)].