Linear second order differential equations in Hilbert spaces. The Cauchy problem and asymptotic behaviour for large time. (English) Zbl 0563.35041

The author considers existence, uniqueness, regularity, and asymptotic behaviour of the solution of the problem \(u''(t)+A(t)u(t)=G(t)+P(t)\); \(u(0)=u_ 0\); \(u'(0)=u_ 1\) assuming (i) \(A(t)\in L(V,V^*)\) a.e. \((0,+\infty)\); \(\| u\|^ 2\leq \lambda (A(t)u,u)\); (ii) the map \(t\to A(t)\) is of bounded variation on the half-line \((0,+\infty)\); (iii) \(G\in L^ 1((0,+\infty),H)\); p is of bounded variation on the half-line \((0,+\infty)\) in \(V^*\) \((V\subset H\subset V^*)\).
Reviewer: R.Salvi


35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
34G10 Linear differential equations in abstract spaces
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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