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

Arosio, Alberto

Arch. Ration. Mech. Anal. 86, 147-180 (1984).

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

MSC:

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)

Keywords:

existence; uniqueness; regularity; asymptotic behaviour

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