BERNARDI, Enrico; BOVE, Antonio; PETKOV, Vesselin

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BERNARDI, Enrico
Matemates

BOVE, Antonio

PETKOV, Vesselin
Institut de Mathématiques de Bordeaux [IMB]

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Article de revue

Ce document a été publié dans

Journal of Hyperbolic Differential Equations. 2015-09, vol. 12, n° 3, p. 535-579.

World Scientific Publishing

Résumé en anglais

We study a class of third order hyperbolic operators $P$ in $G = \{(t, x):0 \leq t \leq T, x \in U \Subset \R^{n}\}$ with triple characteristics at $\rho = (0, x_0, \xi), \xi \in \R^n \setminus \{0\}$. We consider the case when the fundamental matrix of the principal symbol of $P$ at $\rho$ has a couple of non-vanishing real eigenvalues. Such operators are called {\it effectively hyperbolic}. V. Ivrii introduced the conjecture that every effectively hyperbolic operator is {\it strongly hyperbolic}, that is the Cauchy problem for $P + Q$ is locally well posed for any lower order terms $Q$. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in $G$ could have triple characteristics in $G$ only for $t = 0$ or for $t = T$. We prove that the operators in our class are strongly hyperbolic if $T$ is small enough. Our proof is based on energy estimates with a loss of regularity.< Réduire

Mots clés en anglais

Cauchy Problem

Effectively Hyperbolic Operators

Triple Characteristics

Energy estimates

URI

https://oskar-bordeaux.fr/handle/20.500.12278/189393

Project ANR

Opérateurs non-autoadjoints, analyse semiclassique et problèmes d'évolution - ANR-11-BS01-0019

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251