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Singular hyperbolic systems. VI: Asymptotic analysis for Fuchsian hyperbolic equations in Gevrey classes. (English) Zbl 0621.35061
[For Part V see ibid. 36, 449-473 (1984; Zbl 0526.35017).]
The paper deals with a class of Fuchsian hyperbolic operators of the form $P=(t\partial _ t)^ m+\sum _{j+| \alpha | \leq m,j<m}t^{p_{j,\alpha}}a_{j,\alpha}(t,x)(t\partial _ t)^ j\partial _ x^{\alpha},$ where (t,x)$$\in [0,T]\times {\mathbb{R}}^ n$$, $$p_{j,\alpha}\in \{0,1,2,...\}$$ and $$a_{j,\alpha}(t,x)\in C^{\infty}([0,T],{\mathcal S}^{\{s\}}({\mathbb{R}}^ n))$$. Here, $${\mathcal E}^{\{s\}}({\mathbb{R}}^ n)$$ denotes the set of all Gevrey functions on $${\mathbb{R}}^ n$$ of class $$\{$$ $$s\}$$. Under a suitable hyperbolicity, the irregularity index $$\sigma$$ ($$\geq 1)$$ is defined for P. The operator $L=(t\partial _ t)^ 2-t^{2p_ 1}\partial ^ 2_{x_ 1}- t^{2p_ 2}\partial ^ 2_{x_ 2}+t^{q_ 1}a_ 1(t,x)\partial _{x_ 1}+t^{q_ 2}a_ 2(t,x)\partial _{x_ 2}+b(t,x)(t\partial _ t)+c(t,x)$ is a typical example, and in this case $$\sigma$$ is given by $$\sigma =\max \{1,(2p_ 1-q_ 1)/p_ 1$$, $$(2p_ 2-q_ 2)/p_ 2\}$$. Under $$1<s<\sigma /(\sigma -1)$$ and a suitable assumption on the characteristic exponents of P, the following (1) and (2) are established: (1) the unique solvability of $$Pu=f$$ in $$C^{\infty}([0,T],{\mathcal E}^{\{s\}}({\mathbb{R}}^ n))$$, and (2) the asymptotic expansion (as $$t\to +0)$$ of solutions of $$Pu=0$$ in $$C^{\infty}((0,T),{\mathcal E}^{\{s\}}({\mathbb{R}}^ n))$$. In the case $$\sigma =1$$, the results (1) and (2) with $${\mathcal E}^{\{s\}}({\mathbb{R}}^ n)$$ replaced by $${\mathcal E}({\mathbb{R}}^ n)$$ were already obtained in Part III [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 465-507 (1980; Zbl 0463.35053)] and Part V (loc. cit.).

##### MSC:
 35L30 Initial value problems for higher-order hyperbolic equations 35C20 Asymptotic expansions of solutions to PDEs
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