Linear second order differential equations in Hilbert spaces. The Cauchy problem and asymptotic behaviour for large time (Q1057422)

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 \(V^*\) \((V\subset H\subset V^*)\).