Hirosawa, Fumihiko; Reissig, Michael Well-posedness in Sobolev spaces for second-order strictly hyperbolic equations with nondifferentiable oscillating coefficients. (English) Zbl 1047.35079 Ann. Global Anal. Geom. 25, No. 2, 99-119 (2004). Summary: The goal of this paper is to study well-posedness to strictly hyperbolic Cauchy problems with non-Lipschitz coefficients with low regularity with respect to time and smooth dependence with respect to space variables. The non-Lipschitz condition is described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posedness we propose a refined regularizing technique. Two steps of diagonalization procedure basing on suitable zones of the phase space and corresponding nonstandard symbol classes allow to apply a transformation corresponding to the effect of loss of derivatives. Finally, the application of sharp Gårding’s inequality allows to derive a suitable energy estimate. From this estimate we conclude a result about \(C^{\infty}\) well-posedness of the Cauchy problem. Cited in 8 Documents MSC: 35L15 Initial value problems for second-order hyperbolic equations 35L80 Degenerate hyperbolic equations 35S05 Pseudodifferential operators as generalizations of partial differential operators 35B65 Smoothness and regularity of solutions to PDEs Keywords:sharp Garding’s inequality; pseudo-differential operators; diagonalization procedure PDFBibTeX XMLCite \textit{F. Hirosawa} and \textit{M. Reissig}, Ann. Global Anal. Geom. 25, No. 2, 99--119 (2004; Zbl 1047.35079) Full Text: DOI