an:04006832
Zbl 0621.35061
Tahara, Hidetoshi
Singular hyperbolic systems. VI: Asymptotic analysis for Fuchsian hyperbolic equations in Gevrey classes
EN
J. Math. Soc. Japan 39, 551-580 (1987).
00152186
1987
j
35L30 35C20
Fuchsian hyperbolic operators; Gevrey functions; irregularity; unique solvability; asymptotic expansion
[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.).
Zbl 0526.35017; Zbl 0463.35053