Abstract
In this paper we deal with a 2-evolution Cauchy problem coming from the Euler–Bernoulli model for vibrating beams and plates. The leading coefficient, corresponding to the modulus of elasticity, is time-dependent and may vanish at \(t=0\). We prove a well-posedness result in the scale of Sobolev spaces using a \(C^1\)-approach, in this way we have \(H^\infty \) well-posedness with an (at most) finite loss of regularity. We take special interest in the space and time-dependence of a complex coefficient of the extended principle part, related to the shear force, and in the assumptions we pose on that coefficient in order to get \(H^\infty \) well-posedness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ascanelli, A., Boiti, C., Zanghirati, L.: Well-posedness of the Cauchy problem for \(p\)-evolution equations. J. Differ. Equ. 253, 2765–2795 (2012)
Ascanelli, A., Cicognani, M.: Gevrey solutions for a vibrating beam equation. Rend. Semin. Mat. Univ. Politec. Torino 67, 151–164 (2009)
Ascanelli, A., Cicognani, M., Colombini, F.: The global Cauchy problem for a vibrating beam equation. J. Differ. Equ. 247, 1440–1451 (2009)
Cicognani, M., Colombini, F.: The Cauchy problem for \(p\)-evolution equations. Trans. Am. Math. Soc. 362, 4853–4869 (2010)
Herrmann, T.: \(H^\infty \) well-posedness for degenerate \(p\)-evolution operators. Dissertation. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-99818 (2012)
Herrmann, T., Reissig, M.: A sharp result for time-dependent degenerate \(p\)-evolution Cauchy problems of second order. In preparation
Herrmann, T., Reissig, M., Yagdjian, K.: \(H^\infty \) well-posedness for degenerate \(p\)-evolution models of higher order with time-dependent coefficients. To appear
Ichinose, W.: Some remarks on the Cauchy problem for Schrödinger type equations. Osaka J. Math. 21, 565–581 (1984)
Ichinose, W.: The Cauchy problem for Schrödinger type equations with variable coefficients. Osaka J. Math. 24, 853–886 (1987)
Kajitani, K., Baba, A.: The Cauchy problem for Schrödinger type equations. Bull. Sci. Math. 119, 459–473 (1995)
Kubo, A., Reissig, M.: Construction of parametrix to strictly hyperbolic Cauchy problems with fast oscillations in non-Lipschitz coefficients. Commun. Partial Differ. Equ. 28, 1471–1502 (2003)
Mizohata, S.: On the Cauchy problem. Notes and Reports in Mathematics in Science and Engineering, vol. 3. Academic Press, New York (1985)
Reissig, M., Yagdjian, K.: Weakly hyperbolic equation with fast oscillating coefficients. Osaka J. Math. 36, 437–464 (1999)
Acknowledgments
The main part of this paper was obtained during the stays of the second author at the Università di Bologna in 2009 and 2010. The authors thank the Dipartimento di Matematica for this great opportunity.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cicognani, M., Herrmann, T. \(H^\infty \) well-posedness for a 2-evolution Cauchy problem with complex coefficients. J. Pseudo-Differ. Oper. Appl. 4, 63–90 (2013). https://doi.org/10.1007/s11868-013-0062-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-013-0062-4