Asymptotic Decay for Some Differential Systems with Fading Memory

Abstract

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.

Exponential and polynomial decay for first order linear Volterra evolution equations

“…Before proceeding with the proof, we discuss the main result in comparison with the necessary conditions obtained by Fabrizio and Polidoro (see [9]) and mentioned in the introduction. Indeed, in [9], for the particular case of the concrete formulation (1.2), the authors proved (by means of Laplace transform methods) that if the solution u satisfies…”

“…Again, in [9] a result concerning the necessity of such a condition for polynomial decay is established. We will return to these issues with some detail later in Section 3.…”

“…Concerning polynomial stability, we first recall the necessary condition of [9]: defining p 0 = sup r ≥ 0 :…”

“…Nevertheless, our aim is to take advantage of the Volterra finite time delay in order to study the stability under more general assumptions than (1.6). To this purpose, Fabrizio and Polidoro in [9] make use of the so-called exponential decay property…”

Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

“…We also mention that Fabrizio and Polidoro [22] obtained the exponential decay result under the conditions that…”