Abstract
We construct the fundamental solution for a weakly hyperbolic operator satisfying an intermediate condition between effective hyperbolicity and the Levi condition. By the fundamental solution, we obtain the well-posedness in C ⋅ of the Cauchy problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Agliardi and M. Cicognani, The Cauchy problem for a class of Kovalevskian pseudo-differential operators. Proc. Amer. Math. Soc. 132 (2004), 841–845.
A. Ascanelli and M. Cicognani, Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems. J.Differential Equations 217 (2005) n. 2, 305–340.
M. Cicognani, The Cauchy problem for strictly hyperbolic operators with nonabsolutely continuous coefficients. Tsukuba J. Math. 27 (2003), 1–12.
M. Cicognani, Coefficients with unbounded derivatives in hyperbolic equations. Math. Nachr. 277 (2004), 1–16.
F. Colombini, H. Ishida and N. Orrù, On the Cauchy problem for finitely degenerate hyperbolic equations of second order.Ark. Mat. 38 (2000), 223–230.
F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on Time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983), 291–312.
F. Colombini and T. Nishitani, On finitely degenerate hyperbolic operators of second order. Osaka J. Math. 41 (2004) 4, 933–947.
V. Ja. Ivriǐ, Conditions for correctness in Gevrey classes of the Cauchy problem for hyperbolic operators with characteristics of variable multiplicity (Russian). Sibirsk. Mat. Ž. 17 (1976), 1256–1270.
H. Kumano-Go, Pseudo-differential operators, The MIT Press, Cambridge, Massachusetts, and London, England, 1981.
T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second order. Comm. Partial Differential Equations 5 (1980), 1273–1296.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Ascanelli, A., Cicognani, M. (2006). The Fundamental Solution for a Second Order Weakly Hyperbolic Cauchy problem. In: Padula, M., Zanghirati, L. (eds) Hyperbolic Problems and Regularity Questions. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7451-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7451-8_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7450-1
Online ISBN: 978-3-7643-7451-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)