Decay rates for semilinear viscoelastic systems in weighted spaces (Q2905739)

The second-order hyperbolic system with dissipation NEWLINE\[NEWLINEu_{tt}-\sum_{j,k=1}^N B^{jk}u_{x_jx_k}+\sum_{j,k=1}^N K^{jk}*u_{x_jx_k}+Lu_t=f(u),\;x=(x_1,...,x_N)\in \mathbb{R}^N,\;t\geq 0;NEWLINE\]NEWLINE with exponentially decaying \(g\) and \(f(u)=-|u|^{p-1}u,\;p\geq 1\), is considered. For all initial data \((u_0,u_1)\in (H^{s+1}(\mathbb{R}^N))\cap L^{1,\gamma}(\mathbb{R}^N))\) with \(\gamma\in [0,1],\) the authors derive decay estimates both for dissipative structures or regularity-loss type models. Treating the Fourier transform in the low frequency region the optimal decay results are obtained.