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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Cauchy problem for hyperbolic operators with variable multiple characteristics

Author: Kazuhiro Yamamoto
Journal: Proc. Amer. Math. Soc. 72 (1978), 109-116
MSC: Primary 35L30
MathSciNet review: 503542
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Abstract: Let $ P(t,x,{D_t},{D_x})$ be a hyperbolic differential operator with the principal symbol $ {p_m}(t,x,\tau ,\xi )$. We assume that $ {P_m}$ is denoted by $ \Pi _{j = 1}^s{(\tau - {\lambda _j})^{{m_j}}}\Pi _{j = s + 1}^{m - N + s}(\tau - {\lambda _j})$ and $ ({\lambda _i} - {\lambda _j})(t,x,\xi ) \ne 0$ if $ (i,j) \ne (k,m - N + k)\;(k = 1, \ldots ,s)$, where $ N = \Sigma _{j = 1}^s{m_j}$ and $ {\lambda _j}(t,x,\xi ) \in {C^\infty }([0,T] \times {R^n} \times ({R^n}\backslash 0))$. Under a generalized condition of E. E. Levi, we shall show that the Cauchy problem $ Pu = f$ in $ [0,T] \times {R^n},D_t^j{u_{\vert t = 0}} = {g_j}(j = 1, \ldots ,m - 1)$ is well posed. When $ {m_j} = 1(j = 1, \ldots ,s)$, our result coincides those of Ohya and Petkov.

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  • [1] Keiichiro Kitagawa and Takashi Sadamatsu, Sur une condition suffisante pour que le problème de Cauchy faiblement hyperbolique soit bien posé. Cas de multiplicité de caractéristiques au plus triple, J. Math. Kyoto Univ. 17 (1977), no. 3, 465–499 (French). MR 0606231,
  • [2] Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580
  • [3] Y. Ohya, Le problème de Cauchy à caractéristiques multiples, Ann. Scuola Norm. Sup. Pisa (to appear).
  • [4] Veselin M. Petkov, The Cauchy problem for a certain class of non-strictly hyperbolic equations with double characteristics, Serdica 1 (1975), no. 3, 372–380 (Russian). MR 0412621

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