Decay rates of semilinear viscoelastic systems in weighted spaces

Abstract

In this paper we consider a class of second-order hyperbolic system which describe viscoelastic materials. If we take the initial data (u0,u1)∈ (Hs+1 (ℝN) ∩ L1,γ(ℝN)) X (Hs(ℝN) ∩ L1,γ(ℝN)) with γ∈ [0,1], then we can derive faster decay estimates than those known before for both dissipative structure or regularity-loss type models. To this end, we will first transform our problem into Fourier space, then, by using the pointwise estimate derived in [4] combined with a device to treat the Fourier transform in the low frequency region, we succeed in proving the optimal decay results to the solutions of our problem. Finally, we use these decay estimates of the linear problem combined with the weighted energy method introduced by Todorova and Yordanov to tackle a semilinear problem.