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Abstract linear hyperbolic equations and asymptotic equivalence as \(t\to +\infty\). (English) Zbl 0564.35065

Let us consider the Cauchy problem for the second order evolution equation (1) \(u''+A(t)u=0\) a.e. in (0,\(\infty)\); (2) \(u(0)=u_ 0\); (3) \(u'(0)=u_ 1\), where for a.e. \(t>0\), A(t) is a self-adjoint positive operator (generally unbounded), defined on D(A(t))\(\subset H\) with values in H, H a Hilbert space. The inverse \(A^{-1}(t)\) is assumed to be defined everywhere.
The paper essentially concerns with two problems: 1) the existence, uniqueness and regularity of the solution of problem (1)-(3), 2) the asymptotic equivalence at \(t=\infty\) of two equations of type (1). There is a presentation of comparative results obtained by the author in some earlier papers [see J. Differ. Equations 39, 291-309 (1981; Zbl 0472.35056); Arch. Ration. Mech. Anal. 86, 147-180 (1984); C. R. Acad. Sci., Paris, Sér. I 295, 83-86 (1982; Zbl 0494.34003); Ann. Mat. Pura Appl., IV. Ser. 135, 173-218 (1983; Zbl 0543.35005); Nonlinear Anal., Theory Methods Appl. (to appear); and the author together with S. Spagnolo, Nonlinear partial differential equations and their applications, Coll. de France Semin., Paris 1982-83, Vol. VI, Res. Notes Math. 109, 1-26 (1984)].
Reviewer: C.Simionescu

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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