Wave equation with memory

$u'(t) = Au(t) + \int_{-r}^0 k(s)A_1 u(s) ds + f(t),\quad t\ge 0;\quad u(t) = z(t), \quad t\in [-r,0]$

(where $A : D(A)\subset X \to X$ is a closed operator and $A_1 : D(A)\to X$ is continuous) is proved and applied to get a classical solution of the wave equation with memory effects

$ w_{t t} (t,x) = w_{x x}(t, x) + \int_{-r}^0 k(s) w_{x x} (t + s, x)ds + f(t, x), \quad t\ge 0,\quad x\in [0,l]$

To include also the Dirichlet boundary conditions and to get $C^2$-solutions, $D(A)$ is not supposed to be dense hence A is only a Hille-Yosida operator. The methods used are based on a reduction of the inhomogeneous equation to a homogeneous system of the first order and then on an immersion of $X$ in its extrapolation space, where the regularity and perturbation results of the classical semigroup theory can be applied.