Singular hyperbolic systems. VII: Asymptotic analysis for Fuchsian hyperbolic equations in Gevrey classes (2). (English) Zbl 0702.35148

Tahara, Hidetoshi

Jap. J. Math., New Ser. 15, No. 2, 275-307 (1989).

In Part VI [J. Math. Soc. Japan 39, 551-580 (1987; Zbl 0621.35061)] the author defined the irregularity index \(\sigma\) (\(\geq 1)\) for a class of Fuchsian hyperbolic operators P and he developed the asymptotic analysis for \(Pu=f\) in the space of functions of Gevrey classes of exponent s under the assumption \(1<s<\sigma /(\sigma -1)\). The author investigates the same problem under the assumption \(s=\sigma /(\sigma -1)\). The typical example is as follows: \[ P=(t\partial_ t)^ 2- t^{2p}\partial^ 2_ x+a(t,x)t^ q\partial_ x+b(t,x), \] \(\sigma\) \(=\max \{1,(2p-q)/p\}\), where p and q are positive integers.

Reviewer: Hidetoshi Tahara

MSC:

35L30	Initial value problems for higher-order hyperbolic equations
35L10	Second-order hyperbolic equations
35C20	Asymptotic expansions of solutions to PDEs

Keywords:

irregularity index; Fuchsian hyperbolic operators; asymptotic analysis; Gevrey classes