We study the large time behavior of the solution u to an initial and boundary value problem related to the following integro-differential equationIt is known that, when G ≡ 0 and a > 0, the solution u of (0.1) exponentially decays. Here we prove that, for any non-negative a and for any G ≡ 0, the solution u of the equation (0.1) exponentially decays only if the relaxation kernel G does. In other words, the introduction of the dissipative term related to G does not allow the exponential decay due to the presence of the positive coefficient a. We also prove analogous results for the polynomial decay.