Energetic questions about the impact of a viscoelastic solid against an obstacle (Q1087046)

The authors look for positive solutions u(x,t) of the equation with constant coefficients \[ L_ 1u\equiv [\delta \frac{\partial}{\partial t}(\frac{\partial^ 2}{\partial t^ 2}-\frac{\partial^ 2}{\partial x^ 2})+\frac{\partial^ 2}{\partial t^ 2}-k\frac{\partial^ 2}{\partial x^ 2}]\cdot u(x,t)=f(x,t), \] where \(\epsilon >0\), \(0<k<1\). They extend the method of \textit{L. Amerio} and \textit{G. Prouse} [Rend. Mat., IV. Ser. 8, 563-585 (1975; Zbl 0327.73070)] who studied the case \(\epsilon =0\). They prove the solution to be given in terms of the Riemann function of the operator \(L_ 1\) and the solution of the general Cauchy problem. They give explicite expressions for the energy functional along space-like lines that contains a term connected with the memory of the material.