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Perturbative methods in scales of Banach spaces: applications for Gevrey regularity of solutions to semilinear partial differential equations. (English) Zbl 1178.35139

Summary: The goal of this article is to present new trends in the theory of solutions valued in Sobolev spaces for strictly hyperbolic Cauchy problems of second order with non-Lipschitz coefficients. A very precise relation between oscillating behaviour of coefficients and loss of derivatives of solution is given. Several methods as energy method together with sharp Gårding’s inequality and construction of parametrix are used to get optimal results. Counter-examples complete the article.

MSC:

35H10 Hypoelliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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