Abstract
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.
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References
Agliardi, R., Cicognani, M.: Operators of p-evolution with nonregular coefficients in the time variable. J. Differ. Equ. 202(1), 841–845 (2004). doi:10.1090/S0002-9939-03-07092-8
Agliardi, R., Cicognani, M.: The Cauchy problem for a class of Kovalevskian pseudo-differential operators. Proc. Am. Math. Soc. 132(3), 143–157 (2004). doi:10.1016/j.jde.2004.03.028
Bonet, J., Meise, R., Melikhov, S.M.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. 14(3), 425–444 (2007)
Cicognani, M.: Strictly hyperbolic equations with non regular coefficients with respect to time. Ann. Univ. Ferrara Sez. VII (N.S.) 45, 45–58 (1999)
Cicognani, M., Colombini, F.: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. J. Differ. Equ. 221(1), 143–157 (2006). doi:10.1016/j.jde.2005.06.019
Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola. Norm. Sci. 6(3), 511–559 (1979)
Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77(3), 657–698 (1995). doi:10.1215/S0012-7094-95-07721-7
Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963)
Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Reprint of the 1994 ed., Springer, Berlin (2007)
Jannelli, E.: Regularly hyperbolic systems and Gevrey classes. Ann. Math. Pure Appl. 140(1), 133–145 (1985). doi:10.1007/BF01776846
Kajitani, K.: Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes. J. Math. Kyoto Univ. 23(3), 599–616 (1983)
Kajitani, K., Wakabayashi, S.: Microhyperbolic operators in Gevrey classes. Publ. Res. Inst. Math. Sci. 25(2), 169–221 (1989). doi:10.2977/prims/1195173608
Kajitani, K., Yuzawa, Y.: The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable. Ann. Scuola. Norm. Sci. 5(4), 465–482 (2006)
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)
Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, New York (1973)
Nishitani, T.: Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x. Bull. Sci. Math. 107(2), 113–138 (1983)
Pascu, M.: On the definition of Gelfand-Shilov spaces. Ann. Univ. Buchar. Math. Ser. 1(1), 125–133 (2010)
Pilipović, S., Teofanov, N., Tomić, F.: On a class of ultradifferentiable functions. Novi Sad J. Math. 45(1), 125–142 (2015)
Pilipović, S., Teofanov, N., Tomić, F.: Beyond Gevrey regularity. J. Pseudo Differ. Oper. Appl. 7(1), 113–140 (2016)
Reich, M.: Superposition in Modulation Spaces with Ultradifferentiable Weights. arXiv:1603.08723 [math.FA] (2016). Accessed 28 Nov 2016
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Cicognani, M., Lorenz, D. Strictly hyperbolic equations with coefficients low-regular in time and smooth in space. J. Pseudo-Differ. Oper. Appl. 9, 643–675 (2018). https://doi.org/10.1007/s11868-017-0203-2
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DOI: https://doi.org/10.1007/s11868-017-0203-2
Keywords
- Higher order strictly hyperbolic Cauchy problem
- Modulus of continuity
- Loss of derivatives
- Pseudodifferential operators