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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

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)
Full Text: DOI
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