Polynomial stability without polynomial decay of the relaxation function

We consider a linear viscoclastic problem and prove polynomial asymptotic stability of the steady state. This work improves previous works where it is proved that polynomial decay of solutions to the equilibrium state occurs provided that the relaxation function itself is polynomially decaying to zero. In this paper we will not assume any decay rate of the relaxation function. In case the kernel has some flat zones then we prove polynomial decay of solutions provided that these flat zones are not too big. If the kernel is strictly decreasing then there is no need for this assumption.