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Gevrey well-posedness of an abstract Cauchy problem of weakly hyperbolic type. (English) Zbl 0706.35077
The main result presented in this paper is a global existence theorem for solutions to a general second order Cauchy problem of weakly hyperbolic type which is studied in an abstract Hilbert space setting. More precisely, let (H,$$| \cdot |)$$ be a Hilbert space, $$V\subset H$$ a dense subset, $$B:=(b_ 1,...,b_ n)$$, $$n\in {\mathbb{N}}$$, an n-tuple of commuting densely defined closed linear operators $$b_ i\in {\mathcal L}(V,H).$$
For fixed $$s\in R$$, $$s>1$$, $$B=(b_ 1,...,b_ n)$$ generates the Banach scale of generalized Gevrey spaces $$X^ s_ r(B).$$
Let $$T\in R^+$$, $$I:=[0,T]$$, $$k\in]0,2[$$, $$A\in C^ k(I;{\mathcal L}(V,V'))$$ with $$<A(t)v,w>=<A(t)w,v>$$, v,w$$\in V$$, $$M\in L^ 1(I;{\mathcal L}(V,H))$$, $$f\in L^ 1(I;H)$$ and $$s\in [1,1+(k/2)[.$$
In case $$s>1$$ the additional assumption is made that H possesses a countable basis consisting of common eigenvectors of the operators $$b_ i$$, $$i=1,...,n$$. Under these hypotheses and the weak hyperbolicity condition $$<A(t)v,v>\geq 0$$, $$v\in V$$, the author proves that for sufficiently small $$r_ 0$$ and any $$u_ 0,u_ 1\in X^ s_{r_ 0}(B)$$ there exists a unique smooth solution u: $$I\to X^ s_{\bar r}$$ for some $$\bar r\in]0,r_ 0]$$ depending on T to the Cauchy problem $(C)\quad u''+[A(t)+M(t)]u=f(t),\quad t\in I,\quad u(0)=u_ 0,\quad u'(0)=u_ 1$ an analogous existence and uniqueness result for strongly hyperbolic equations is also derived.
The proof depends in an essential way on certain a priori estimates for the “Gevrey-type infinite order energy function” of the solution to (C); the author first proves existence of solutions $$u_ N$$ to a sequence of approximating Cauchy problems $$(C_ N)$$ and then by applying the a priori estimates to these solutions he obtains a uniform bound for the infinite order energy functions of $$u_ N$$; a compactness argument yields the desired result. As an application of his abstract theorems the author re-proves some global and local existence results found earlier by Jannelli and Nichitani in a different approach to the Cauchy problem for weakly and strongly hyperbolic second order equations [see E. Jannelli, J. Math. Kyoto Univ. 24, 763-778 (1984; Zbl 0582.35070), and T. Nichitani, Bull. Sci. Math., II. Ser. 107, 113-138 (1983; Zbl 0536.35042)]. Apart from this some applications to certain concrete differential operators are also considered.
Reviewer: H.-J.Böttger

##### MSC:
 35L15 Initial value problems for second-order hyperbolic equations 34G10 Linear differential equations in abstract spaces 35B45 A priori estimates in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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