# zbMATH — the first resource for mathematics

Cauchy problems for Fuchsian hyperbolic equations in spaces of functions of Gevrey classes. (English) Zbl 0586.35060
This note deals with the Cauchy problem $t^ k \partial^ m_ t u+\sum t^{p(j,\alpha)} a_{j,\alpha} \partial^ j_ t \partial_ x^{\alpha} u=f(t,x),\quad \partial^ i_ t u(0,x)=u_ i(x),$ i$$=0,1,...,m-k-1$$, $$0\leq t\leq T$$, $$x\in {\mathbb{R}}^ n$$, where m, k are integers, $$0\leq k<m$$, $$0\leq p(j,\alpha)$$ suitable integers, $$j+| \alpha | \leq m$$, $$j<m$$, with suitable hyperbolicity conditions. The problem is well posed in spaces of $$C^{\infty}$$ functions and in spaces of functions of Gevrey classes. Moreover the solution has finite propagation speed.
Reviewer: P.Jeanquartier

##### MSC:
 35L30 Initial value problems for higher-order hyperbolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 58J47 Propagation of singularities; initial value problems on manifolds
Full Text:
##### References:
 [1] Igari, K.: An admissible data class of the Cauchy problem for non-strictly hyperbolic operators. J. Math. Kyoto Univ., 21, 351-373 (1981). · Zbl 0479.35058 [2] Ivrii, V. Ja.: Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes. Sib. Mat. Zh., 17, 1256-1270 (1976). · Zbl 0352.35060 [3] Komatsu, H.: Ultradistributions, I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 20, 25-105 (1973). · Zbl 0258.46039 [4] Mandai, T.: Necessary conditions for well-posedness of the flat Cauchy problem and the regularity loss of solutions. Publ. RIMS, Kyoto Univ., 19, 145-168 (1983). · Zbl 0513.35045 · doi:10.2977/prims/1195182981 [5] Tahara, H.: Cauchy problems for Fuchsian hyperbolic partial differential equations. Proc. Japan Acad., 54A, 92-96 (1978). · Zbl 0403.35062 · doi:10.3792/pjaa.54.92 [6] Tahara, H.: Singular hyperbolic systems, III. On the Cauchy problem for Fuchsian hyperbolic partial differential equations. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 27, 465-507 (1980). · Zbl 0463.35053 [7] Uryu, H.: Conditions for well-posedness in Gevrey classes of the Cauchy problems for Fuchsian hyperbolic operators (preprint). · Zbl 0585.35060 · doi:10.2977/prims/1195179627
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.