Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity (Q2902746)

The paper is devoted to the study of the sharp decay estimates of solutions for quasilinear hyperbolic systems of viscoelasticity NEWLINE\[NEWLINE\begin{aligned} &u_{tt}-\sum_jb^j(\partial_xu)_{x_j}+\sum_{j,k}K^{jk}*u_{x_jx_k}+Lu_t=0,\\ & u(x,0)=u_0(x),\;u_t(x,0)=u_1(x) \end{aligned}NEWLINE\]NEWLINE with an unknown vector function \(u\) of \(x=(x_1,\dots,x_n)\in \mathbb{R}^n,\;n\geq 1\), and smooth symmetric real matrix functions \(K^{jk}(t),\;t\geq 0.\) The authors develop the time-weighted energy method enabling them to find a decay estimate of solutions for small initial data in \(L^2\), provided that \(n\geq 2\) . Applying the semigroup argument an optimal decay estimate for small initial data in \(L^2\cap L^1\) and for all \(n\geq 1\) is achieved, too.