The Cauchy problem for hyperbolic operators with variable multiple characteristics

Yamamoto, Kazuhiro

doi:10.1090/S0002-9939-1978-0503542-1

The Cauchy problem for hyperbolic operators with variable multiple characteristics
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by Kazuhiro Yamamoto PDF

Proc. Amer. Math. Soc. 72 (1978), 109-116 Request permission

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 $[0,T] \times {R^n},D_t^j{u_{|t = 0}} = {g_j}(j = 1, \ldots ,m - 1)$ is well posed. When ${m_j} = 1(j = 1, \ldots ,s)$, our result coincides those of Ohya and Petkov.

References

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 606231, DOI 10.1215/kjm/1250522710
Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580

Le problème de Cauchy à caractéristiques multiples

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 412621