Global solutions and decay property with regularity-loss for quasi-linear hyperbolic systems with dissipation (Q2836509)

Under assumptions of sufficient smallness and regularity of the initial data the author proves existence, uniqueness and polynomial decay of a solution of the initial value problem to the hyperbolic system NEWLINE\[NEWLINE u_{tt}-\sum_{j=1}^nb^j(\partial _xu)_{x_j}+(1-\Delta )^{-1}\left(\sum_{j,k=1}^mK^{jk}\star u_{x_jx_k}+Lu_t\right)=0, NEWLINE\]NEWLINE where \(b^j(v)\: \mathbb{R}^{mn}\rightarrow \mathbb{R}^n\) (\(j=1,\dots, n\)) are smooth functions, \(K^{jk}\) (\(K^{jk}=K^{kj}\)) are smooth functions of \(t\) and \(L\) is an \(m\times m\) real symmetric constant matrix.