Maximal regularity results for a second order integro-differential equation (Q1898942)

Conditions for the existence, uniqueness and maximal regularity of the strict and strong solutions to the integro-differential equation \[ u'' (t) = \eta Au'(t) + \beta Au(t) + \int^t_0 k(t - s) Au(s) ds + f(t), \quad u(0) = x,\;u'(0) = y,\;t \in [0,T], \tag{1} \] are established, where \(A\) is a linear operator, \(\eta \geq 0\) and \(\beta\) are real constants, the kernel \(k\) is a real function satisfying some assumptions which guarantee, that \(k\) gives a parabolic character of (1). The cases: \(f \equiv 0\) and \(f\) is a continuous known function are investigated separately. The results are proved by Laplace transform methods. Some regularity results concerning the convolution term \(R*f\) are also given.