On the Cauchy problem for hyperbolic operators of second order whose coefficients depend only on the time variable. (English) Zbl 1203.35161

Wakabayashi, Seiichiro

J. Math. Soc. Japan 62, No. 1, 95-133 (2010).

The author deals with hyperbolic operators of second-order whose coefficients depend only on the time variable and gives necessary and sufficient conditions for the Cauchy problem to be \(C^\infty\) well-posed. In particular, he gives a necessary and sufficient condition for \(C^\infty\) well-posedness when the space dimension is equal to 2 and coefficients are real analytic functions of the time variable. He proves that the Cauchy problem for \[ P= D^2_t- \sum a_{j,k}(t) D_{x_j} D_{x_k}+ \sum b_j(t) D_j+ b_0(t) D_t+ c(t) \] is \(C^\infty\) well-posed, if the coefficients satisfy \(\sum a_{j,k}(t)\xi_j \xi_k\geq 0\), and for any \[ t_0\in [0,T]\min_{r\in R_j}|t-\tau|\leq C\sqrt{\sum a_{j,k}(t)\xi_j\xi_k} \] for \((t,\xi)\in U\times\Gamma_j\), where \(U\) is a neighborhood of \(t_0\), \(\Gamma_j\) a conic neiborhood in \(\mathbb{R}^n\), \(R_j= \{(\text{Re\,}\lambda)_+; \lambda\in C,q_j(\lambda, \xi)= 0\), \(\text{Re\,}\lambda\in [t_0- 2\delta, t_0+\delta]\}\). \(q(\lambda,\xi)\) is a real-valued polynomial of \(t\) and is defined by \[ \sum a_{j,k}(t)\xi_j\xi_k= e_j(t,\xi) q(t,\xi),\;(t,\xi)\in U\times\Gamma_j, \] where \(e_j(t,\xi)\) satisfies that \(e_j(t,\xi)\geq 0\) and \(\partial_t e(t, \xi)\leq Ce_j(t,\xi)\), \((t,\xi)\in U\times\Gamma_j\).

Reviewer: K. Kajitani (Ibaraki)

MSC:

35L15	Initial value problems for second-order hyperbolic equations
35L10	Second-order hyperbolic equations

Keywords:

Cauchy problem; hyperbolic; \(C^\infty\)-well-posed

References:

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