Strictly hyperbolic equations with coefficients low-regular in time and smooth in space
- 102 Downloads
We consider the Cauchy problem for strictly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is \(L^2\) well-posed in the case of Lipschitz continuous coefficients in time, \(H^s\) well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of Hölder continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and Hölder coefficients are recovered.
KeywordsHigher order strictly hyperbolic Cauchy problem Modulus of continuity Loss of derivatives Pseudodifferential operators
Mathematics Subject Classification35S05 35L30 47G30
- 9.Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Reprint of the 1994 ed., Springer, Berlin (2007)Google Scholar
- 14.Komatsu, H.: Ultradistributions and hyperbolicity. In: Garnir, H.G. (ed.) Boundary Value Problems for Linear Evolution PartialDifferential Equations: Proceedings of the NATO Advanced StudyInstitute held in Liège, Belgium, September 6–17, 1976, pp. 157–173. Springer Netherlands, Dordrecht (1977)Google Scholar
- 20.Reich, M.: Superposition in Modulation Spaces with Ultradifferentiable Weights. arXiv:1603.08723 [math.FA] (2016). Accessed 28 Nov 2016