DECAY RATES FOR SEMILINEAR VISCOELASTIC SYSTEMS IN WEIGHTED SPACES.

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)∈(Hs+1(ℝN) ∩ L1, γ(ℝN))×(Hs(ℝN) ∩ L1, γ(ℝN)) 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. [ABSTRACT FROM AUTHOR]