Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity
Journées équations aux dérivées partielles (2010), article no. 4, 13 p.

We study a class of third order hyperbolic operators $P$ in $G=\Omega \cap \left\{0\le t\le T\right\},\phantom{\rule{0.222222em}{0ex}}\Omega \subset {ℝ}^{n+1}$ with triple characteristics on $t=0$. We consider the case when the fundamental matrix of the principal symbol for $t=0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t>0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P+Q$ is well posed in $G$ for any lower order terms $Q$.

@article{JEDP_2010____A4_0,
author = {Bernardi, Enrico and Bove, Antonio and Petkov, Vesselin},
title = {Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2010},
doi = {10.5802/jedp.61},
language = {en},
url = {http://www.numdam.org/item/JEDP_2010____A4_0}
}

Bernardi, Enrico; Bove, Antonio; Petkov, Vesselin. Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity. Journées équations aux dérivées partielles (2010), article  no. 4, 13 p. doi : 10.5802/jedp.61. http://www.numdam.org/item/JEDP_2010____A4_0/

[1] E. Bernardi, A. Bove, V. Petkov, The Cauchy problem for effectively hyperbolic operators with triple characteristics, in preparation.

[2] J. M. Bony, Sur l’inégalité de Fefferman-Phong, Séminaire EDP, Ecole Polytechnique, 1998-1999. | Numdam | MR 1721321

[3] J. Chazarain, Opérateurs hyperboliques à caractéristiques de multiplicité constante, Ann. Institut Fourier (Grenoble), 24 (1974), 173-202. | Numdam | MR 390512 | Zbl 0274.35045

[4] H. Flashka and G. Strang, The correctness of the Cauchy problem, Adv. in Math. 6 (1971), 347-379. | MR 279425 | Zbl 0213.37304

[5] L. Hörmander, Cauchy problem for differential operators with double characteristics, J. Analyse Math. 32 (1977), 118-196. | MR 492751 | Zbl 0367.35054

[6] L. Hörmander, Analysis of Linear Partial Differential Operators, III, Springer-Verlag, 1985, Berlin. | MR 781536 | Zbl 0601.35001

[7] V. Ja. Ivrii and V. M. Petkov, Necessary conditions for the Cauchy problem for non-strictly hyperboilic equations to be well posed, Uspehi Mat. Nauk, 29: 5 (1974), 1-70 (in Russian), English translation: Russian Math. Surveys, 29:5 (1974), 3-70. | MR 427843 | Zbl 0312.35049

[8] V. Ivrii, Sufficient conditions for regular and completely regular hyperbolicity, Trudy Moskov Mat. Obsc.,33 (1976), 3-66 (in Russian), English translation: Trans. Moscow Math. Soc. 1 (1978), 165. | MR 492904 | Zbl 0376.35038

[9] N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (a standard type), Publ. RIMS Kyoto Univ. 20 (1984), 551-592. | MR 759681

[10] N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (general case), J. Math. Kyoto Univ. 25 (1985), 727-743. | MR 810976 | Zbl 0613.35046

[11] R. Melrose, The Cauchy problem for effectively hyperbolic operators, Hokkaido Math. J. 12 (1983), 371-391. | MR 725587 | Zbl 0544.35094

[12] T. Nishitani, Local energy integrals for effectively hyperbolic operators, I, II, J. Math. Kyoto Univ. 24 (1984), 623-658 and 659-666. | MR 775976 | Zbl 0589.35078

[13] T. Nishitani, The effectively Cauchy problem in The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, 1505, Springer-Verlag, 1991, pp. 71-167. | MR 1166190

[14] O. A. Oleinik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math. 23 (1970), 569-586. | MR 264227

[15] M. R. Spiegel, J. Liu, Mathematical handbook of formulas and tables, McGraw-Hill, Second Edition, 1999.