On the decay of the energy for systems with memory with indefinite dissipation

In this paper we study the asymptotic behavior of the viscoelastic system with non dissipative kernels. We show that the uniform
decay of the energy depends on the decay of the kernel, the positivity
of the kernel in t = 0 and some smallness condition. That is, if the
kernel g ∈ C^2(R^+) with g(0) > 0, decays exponentially to zero then the
solution decays exponentially to zero. On the other hand, if the kernel
decays polynomially as t^{−p} then the corresponding solutions also decays
polynomially to zero with the same rate of decay