## $L^2$-theory of singular perturbation of hyperbolic equations III Asymptotic expansions of dispersive type

J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 1, Page 199--246.

Uchiyama, KÃ´ichi
$L^2$-theory of singular perturbation of hyperbolic equations III Asymptotic expansions of dispersive type
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Abstract:
We consider Cauchy problems for linear strictly hyperbolic equations of order $l$ with a small parameter $Îµ \in (0, Îµ_0 ]$ : % \begin{eqnarray} &\{ &\hspace{-3mm} (i Îµ )^{l-m} L (t,x,D_t , D_x; Îµ)+ M(t,x,D_t , D_x ;Îµ) \} u(t,x; Îµ) \ && =f(t,x;Îµ) \nonumber \ &&\mbox{for} (t,x) \in (0,T) Ã \mbox{\boldmath $R$ }_x^d, \nonumber \end{eqnarray} % $$D_t^j u(0,x; Îµ ) = g_j (x;Îµ) j=0,1,2, \ldots, l-1 \label{eqn: 0.2}$$ % 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 $(t,x,Îµ) \in [0,\infty) Ã \mbox{\boldmath R }^d Ã [0,Îµ_0]$. 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 $Îµ$ 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.

Mathematics Subject Classification (1991): Primary 35L30; Secondary 35B25, 35C20, 81Q20
Mathematical Reviews Number: MR1414625

Received: 1994-04-11