Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

Abstract

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis 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-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Garding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C∞well-posedness of the Cauchy problem.