The authors consider a strongly damped wave equation with linear memory term:
on with on , where for and . A variety of conditions are imposed on the data , , . After some preparations, (1) is transformed into a system:
System (2) is then cast into a functional frame. With , the Dirichlet Laplacian, fractional powerspaces , are introduced for in the usual way. A weighted Hilbert space is then defined via . The phase space for (2) then is .
A function is called translation invariant if the closure of all translates , is compact in . Finally, a function space is defined, consisting of the with norm .