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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
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