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タイトル: $L^2$-theory of singular perturbation of hyperbolic equations III Asymptotic expansions of dispersive type
著者: Uchiyama, Koichi
発行日: 1996年
出版者: Graduate School of Mathematical Sciences, The University of Tokyo
掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 3 (1996), No. 1, Page 199-246
抄録: We consider Cauchy problems for linear strictly hyperbolic equations of order $l$ with a small parameter $ε \in (0, ε_0 ]$ : % \begin{eqnarray} &\{ &\hspace{-3mm} (i ε )^{l-m} L (t,x,D_t , D_x; ε)+ M(t,x,D_t , D_x ;ε) \} u(t,x; ε) \\ && =f(t,x;ε) \nonumber \\ &&\mbox{for} (t,x) \in (0,T) × \mbox{\boldmath $R$ }_x^d, \nonumber \end{eqnarray} % \begin{equation} D_t^j u(0,x; ε ) = g_j (x;ε) j=0,1,2, \ldots, l-1 \label{eqn: 0.2} \end{equation} % where $L$ and $M$ are linear strictly hyperbolic operators of order $l$ and $m$ \((l = m+1\) or $m+2$) with $C^\infty$ bounded derivatives with respect to \((t,x,ε) \in [0,\infty) × \mbox{\boldmath $R$ }^d × [0,ε_0]\). The aim of this paper is to give \(C^{\infty}\) asymptotic expansions of solutions to singularly perturbed Cauchy problems of this type, when the characteristic roots of $L$ and $M$ satisfy the separation conditions. The points are to construct formal solutions (Proposition 5.3, 5.4), consisting of the regular part and the singular one (correction part of dispersive type) expressed by Maslov's canonical operators, and to give the error estimates in order to obtain asymptotic expansions with respect to $ε$ in the sense of arbitrarily higher order differentiability norms (Theorem 6.1, 6.2), when the supports of $f$ and $g_j$'s are contained in fixed compact sets.
URI: http://hdl.handle.net/2261/1541
ISSN: 13405705
出現カテゴリ:Journal of Mathematical Sciences, the University of Tokyo
Journal of Mathematical Sciences, the University of Tokyo

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