Asymptotic Decay for Some Differential Systems with Fading Memory

We study the large time behavior of the solution u to an initial and boundary value problem related to the following integro-differential equation uₜₜ = G₀ Δ u + ∫₀ᵗ G'(t-s) Δ u(x, s)\, ds - a uₜ (0.1) where G 0 , a are real constant coefficients, G 0 > 0, a S 0 and G\,' ∈ L¹(^ + ) ∩ L²(^ + ), G\,' ≤ 0 . It is known that, when G ' L 0 and a > 0, the solution u of (0.1) exponentially decays. Here we prove that, for any nonnegative a and for any G ' ≡ 0 , the solution u of the Eq. (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.