The lack of exponential stability for a class of second-order systems with memory

We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equation u(t) + Au(t) + integral(infinity)(0) mu(s)A(gamma)[ u(t) - u(t - s)] ds = 0, where the operator A is strictly positive self-adjoint with A(-1) not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter gamma is an element of R. In particular, we show that S(t) is not exponentially stable when gamma not equal 1.