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Singular hyperbolic systems. VIII: On the well-posedness in Gevrey classes for Fuchsian hyperbolic equations. (English) Zbl 0774.35044
Summary: [For the former parts see: the author, Jap. J. Math., New Ser. 15, No. 2, 275-307 (1989; Zbl 0702.35148).]
The paper discusses the well-posedness problem in Gevrey classes for Fuchsian type partial differential equations $$Pu=f$$ with $$P$$ being of the form $P=(t\partial_ t)^ m+\sum_{j+|\alpha|\leq m,j<m}t^{l(j,\alpha)}a_{j,\alpha}(t,x)(t\partial_ t)^ j\partial^ \alpha_ x.$ The main subject is to investigate the difference between the following two assertions: (A) $$Pu=f$$ is wellposed in $$C^ \infty([0,T]$$, $${\mathcal E}^{\{s\}}(\mathbb{R}^ n))$$, and (B) $$Pu=f$$ is wellposed in $${\mathcal E}^{\{s\}}([0,T]\times\mathbb{R}^ n)$$, where $${\mathcal E}^{\{s\}}(\Omega)$$ denotes the space of all Gevrey functions of class $$\{s\}$$ on $$\Omega$$. The author’s motivation comes from the following example: in the case $$P=(t\partial_ t+1)^ 2-t\partial^ 2_ x$$, (A) is true for all $$s>1$$, but (B) is not true for any $$s>1$$.

##### MSC:
 35L25 Higher-order hyperbolic equations 35L80 Degenerate hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs 35L30 Initial value problems for higher-order hyperbolic equations 35C20 Asymptotic expansions of solutions to PDEs
well-posedness