Global solutions to quasi-linear hyperbolic systems of viscoelasticity (Q542656)

The authors deal with the initial value problem \[ \begin{aligned} &u_{tt}-\sum_{j=1}^n b^j(\partial_xu)_{x_j}+\sum_{j,k=1}^n K^{jk}*u_{x_jx_k}+Lu_t=0,\\ &u(x,0)=u_0(x),\;\;u_t(x,0)=u_1(x),\quad x\in \mathbb R^n \end{aligned} \] with smooth \(m\)-vector functions \(b^j\), an unknown \(m\)-vector function \(u\), smooth \(m \times m\) real matrix functions \(K^{jk}\) and the symmetric \(m\times m\) matrix \(L\). The system is assumed to have a free energy \(\varphi(v)\) satisfying \(b^j(v)=D_{v_j}\varphi(v)\), where \(D_{v_j}\varphi(v)\) denotes the Fréchet derivative of \(\varphi(v)\) with respect to \(v_j\). The system has dissipative properties due to a memory and a damping term. It is proved that the solution exists globally in time, provided that the initial data are sufficiently small. Moreover the solution converges to zero as time tends to infinity. The crucial point of the proof are uniform a priori estimates of solutions by using an energy method.