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

摘要

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 G?rding's inequality allows to derive a suitable energy estimate. From this estimate we conclude a result about C∞ well-posedness of the Cauchy problem.