On the Cauchy problem for a class of hyperbolic operators with triple characteristics
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On the Cauchy problem for a class of hyperbolic operators with triple characteristics Annamaria Barbagallo1 · Vincenzo Esposito1 Received: 9 July 2020 / Accepted: 1 October 2020 © The Author(s) 2020
Abstract The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, an existence result for the Cauchy problem is obtained. Keywords Cauchy problem · Hyperbolic equations · Pseudodifferential operators Mathematics Subject Classification 35B45 · 35S05 · 35L30
1 Introduction In the past, many authors studied widely hyperbolic operators with double characteristics, both in the case when there is no transition between different types on the set where the principal symbol vanishes of order 2 (see for instance [5,8] for a general survey) and when there is transition (see [1–4]). The operators are called effectively hyperbolic if the propagation cone C is transversal to the manifold of multiple points (see [8]). Moreover, if this occurs and lower order terms satisfy a generic Ivrii-Petkov vanishing condition, we have well posedness in C ∞ (see [7]). The aim of the paper is to analyze the following class of operators with triple characteristics P(x0 , D) = Dx30 − (Dx21 + x12 Dx22 )Dx0 − bx13 Dx32 , in Ω =]0, +∞[×R2 , where Dx j = 1i ∂x j , j = 0, 1, 2, under hyperbolicity assumptions, namely |b| ≤ 23 . Such a class of operators has been considered in [6], for example operators whose propagation cone is not transversal to the triple characteristic manifold. The authors
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Annamaria Barbagallo [email protected] Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II, Via Cinthia - Monte S. Angelo, 80126 Naples, Italy
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A. Barbagallo, V. Esposito
prove a well posedness result in the Gevrey category for a simple hyperbolic operator with triple characteristics and whose propagation cone is not transversal to the triple manifold. Furthermore they estimate the precise Gevrey threshold, by exhibiting a special class of solutions, through which we can violate weak necessary solvability conditions. More precisely, let x = (x0 , x ) where x = (x1 , x2 ), let ξ = (ξ0 , ξ ), where ξ = (ξ1 , ξ2 ). In [6], the authors study the well posedness of the following Cauchy problem
Pu = 0, in Ω =]0, +∞[×R2 , Dx j u(0, x ) = φ j (x ), j = 0, 1, 2,
with φ j (x ) ∈ γ (s) (R2 ), j = 0, 1, 2, where γ (s) (R2 ) is the Gevrey s class. They obtained that the Cauchy problem for P is well posed in the Gevrey 2 class assuming 4 2 such that the . Moreover, if s > 2, it is possible to choose b ∈ 0, √ that b2 < 27 3 3 Cauchy problem for P is not locally solvable at the origin in the Gevrey s class. In this paper, instead, we investigate on the well posedness of the Cauchy problem
Pu = f , in Ω, Dx j u(0, x ) = 0, j = 0, 1, 2,
(1)
with f ∈ H r (Ω), in the Sobolev spaces, obtaining an existence result for solutions. Let us set Q = −∂x30 + ∂x21 + x12
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