Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

TL;DR

This paper proves that a class of third order effectively hyperbolic operators with triple characteristics are strongly hyperbolic for small enough time intervals, extending the understanding of well-posedness in hyperbolic PDEs.

Contribution

It establishes strong hyperbolicity for a new class of effectively hyperbolic operators with triple characteristics, using energy estimates with regularity loss.

Findings

Operators are strongly hyperbolic if the time interval is sufficiently small.

The proof employs energy estimates with a loss of regularity.

Extends results to operators with triple characteristics not admitting factorization.

Abstract

We study a class of third order hyperbolic operators $P$ in $G = \{(t, x):0 \leq t \leq T, x \in U \Subset {\mathbb R}^{n}\}$ with triple characteristics at $\rho = (0, x_0, \xi), \xi \in {\mathbb R}^n \setminus \{0\}$. 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.