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

Abstract

We study a class of third-order hyperbolic operators P in G = {(t, x): 0 ≤ t ≤ T, x ∈ U ⋐ ℝn} with triple characteristics at ρ = (0, x0, ξ), ξ ∈ ℝn ∖{0}. We consider the case when the fundamental matrix of the principal symbol of P at ρ has a couple of non-vanishing real eigenvalues. Such operators are called effectively hyperbolic. Ivrii introduced the conjecture that every effectively hyperbolic operator is 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.