Abstract
A well-known example by Ivrii concerning the operatorP=D 2t −t2lD 2x +atkDx (a≠0), shows that there exists a delicate relation amongl, k and the Gevrey index of well-posedness of the Cauchy problem. In this paper we give a generalization to a class of pseudo-differential operators includingP.
Sunto
Un famoso esempio di Ivrii riguardante l'operatoreP=D 2t −t2lD 2x +atkDx (a≠0), mostra che c'è una relazione sottile tral, k e l'indice di Gevrey di buona positura del problema di Cauchy. In questo articolo viene data una generalizzazione ad una classe di operatori pseudodifferenziali che comprendeP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Agliardi, Fourier integral operators of infinite order onD \(_{L^2 }^{\left\{ \sigma \right\}} \)(D \(_{L^2 }^{\left\{ \sigma \right\}} \))with an application to a certain Cauchy problem, Rend. Sem. Mat. Univ. Padova,84 (1990), pp. 73–82.
L. Cattabriga—L. Zanghirati,Fourier integral operators of infinite order on Gevrey spaces—Applications to the Cauchy problem for certain hyperbolic operators,J. Math. Kyoto Univ.,30-1 (1990), pp. 149–192.
K. Igari,An admissible data class of the Cauchy problem for non-stricly hyperbolic operators,J. Math. Kyoto Univ.,21-2 (1981), pp. 351–373.
V. Ya. Ivrii,Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes,Sib. Mat. Zh.,17 (1976), pp. 1256–1270.
V. Ya. Ivrii,Well-posedness in Gevrey classes of the Cuachy problem for nonstrictly hyperbolic operators, Izv. Vyssh. Uchebn. Zaved. Mat.,2 (1978), pp. 26–35.
V. Ya. Ivrii,Linear hyperbolic operators, in Partial Differential Equations IV, Encyclopaedia of Mathematical Sciences, vol. 33 (Eds.:Yu. V. Egorov—M. A. Shubin), Springer-Verlag (1993), pp. 149–235.
H. Kumano-go,Pseudo-Differential Operators, M.I.T. Press (1981).
Y. Ohya,Le problème de Cauchy à caractéristique multiple, Ann. Scu. Norm. Sup.,4 (1977), pp. 757–805.
Y. Ohya,A remark on the Cauchy on the problem of non-strict hyperbolicity, inDevelopments in Partial Differential Equations and Applications to Mathematical Physics (Eds.:G. Buttazzo—G. P. Galdi—L. Zanghirati), Plenum Press (1992), pp. 115–120.
O. A. Oleinik,On the Cauchy problem for weakly hyperbolic equations,C.P.A.M.,23 (1970), pp. 569–586.
T. Sadamatsu,On a necessary condition for the well-posedness of the Cauchy problem for evolution equations. J. Math. Kyoto Univ.,29-2 (1989), pp. 221–231.
K. Shinkai,On the fundamental solution for a degenerate hyperbolic system, Osaka J. Math.,18 (1981), pp. 257–288.
K. Shinkai—K. Taniguchi,Fundamental solution for a degenerate hyperbolic operator in Gevrey classes,Publ. R.I.M.S.,28 (1992), pp. 169–205.
K. A. Yagdzhyan,Pseudodifferential operators with a parameter and the fundamental solution of the Cauchy problem for operators with multiple characteristics, Soviet J. Contem. Math. Anal.,21-4 (1986), pp. 1–29.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Agliardi, R. A generalization of an example by V. Ya. Ivrii. Ann. Univ. Ferrara 39, 93–109 (1993). https://doi.org/10.1007/BF02826132
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02826132