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Well-posedness in Sobolev spaces for second-order strictly hyperbolic equations with nondifferentiable oscillating coefficients. (English) Zbl 1047.35079

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.

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
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