Narrow Results

We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation

Equations with infinite delay commonly face the philosophical objection of being "unphysical", since a memory of infinite duration conflicts with reality. Indeed, besides common sense, experimental observations on concrete physical models tell that effects from the far past cannot possibly influence the current dynamics of a given system. On the other hand, infinite delay arises quite naturally in the mathematical description of several relevant phenomena. In this note, we propose a possible conceptual solution, showing that infinite delay can be recovered as a limiting case of finite delay on a large time-scale, along with a quantitative control of the discrepancy.

We consider a class of second-order hyperbolic systems which describe viscoelastic materials, and we extend the recent results by Dharmawardane and Conti et al. More precisely, for all initial data (u₀, u₁)∈(H^s+1(ℝ^N) ∩ L^{1, γ}(ℝ^N))×(H^s(ℝ^N) ∩ L^{1, γ}(ℝ^N)) with γ∈[0, 1], we derive faster decay estimates for both dissipative structure or regularity-loss type models. To this end, we first transform our problem into Fourier space and then, by using the pointwise estimate derived by Dharmawardane et al., combined with a device to treat the Fourier transform in the low frequency region, we derive optimal decay results for the solutions to our problem. Finally, we use these decay estimates for the linear problem, combined with the weighted energy method introduced by Todorova and Yordanov, and tackle a semilinear problem.