Singular hyperbolic systems. VI: Asymptotic analysis for Fuchsian hyperbolic equations in Gevrey classes
[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.).
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