Complete Volterra integrodifferential equations of the second order unsolved with respect to the higher derivative (Q317626)

The authors study the evolution of dynamical systems with infinite numbers of degrees of freedom and with regard for the relaxation phenomena by considering in the Hilbert space \(H\), the Cauchy problem for the Volterra integro-differential equation of second order \[ \frac{d^2u}{dt^2}+ (F + iG) + Bu + \sum_{k=1}^{m} \int_{0}^{t} G_k(t, s)C_k u(s)ds = f(t), \] \[ u(0) = u^0, \; u'0) = u^1. \] Assume that \(A\) is a bounded operator (\(A \in L(H)\)) and that the coefficients \(F\), \(G\), \(B\) and \(C_k\) are unbounded, non commuting operators specified in their domains of definition are dense in \(H\). Further, the authors assume that ``the domains of definition of these operators are comparable by selecting the classes of equations for which one of the operators can be regarded as the principal operator''. Also they ``consider that the operator \(A\) acts in the scale of spaces \(E^\alpha\) but not in \(H\)''. With this basic set up, they consider the obtained sufficient conditions for the solvability of the considered problem in the following set ups of: (i) low intensity of internal dissipation; (ii) medium intensity of internal dissipation; and (iii) high intensity of internal dissipation.