Decay property of regularity-loss type for solutions in elastic solids with voids

In this paper, we consider the Cauchy problem for a system of elastic solids with voids. First, we show that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in H^s(R)∩L¹(R)$H^s\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$. Furthermore, we prove that by restricting the initial data to be in H^s(R)∩L^1,γ(R)$H^s\left(\mathbb{R}\right)\cap L^{1,\gamma }\left(\mathbb{R}\right)$ and γ∈[0,1]$\gamma \in [0,1]$, we can derive faster decay estimates of the solution. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay rate.