Global existence and decay rate for a coupled degenerate hyperbolic system with dissipation (Q2842074)

This paper deals with the initial-boundary value problem with homogeneous Dirichlet data for a coupled degenerate hyperbolic \(2\times 2\) system in \(\Omega_x\times [0,\infty)\), \(\Omega_x\) being a smooth bounded domain in \(\mathbb{R}^N\). The coefficient in front of the Laplace operator is NEWLINE\[NEWLINE\int_\Omega(|\nabla_x u(x,t)|^2+ |\nabla_x v(x,t)|^2)\,dx,NEWLINE\]NEWLINE while the coefficient in front of \(u_{tt}(v_{tt})\) is \(\rho> 0\), respectively, the dissipative terms are \(\delta u_t(\delta v_t)\), \(\delta> 0\). When either \(\rho\) or the norms of Cauchy data are sufficiently small with respect to \(\delta\), the author proves the existence of a unique global in time \(t\geq 0\) solution \((u,v)\) of the system in an appropriate Sobolev class. Moreover, the optimal decay rate of \((u(t),v(t))\) satisfies the relation \(\| u(t)\|^2_{H^2(\Omega)}+ \| v(t)\|^2_{H^2(\Omega)}\sim (1+t)^{-1}\), \(t\geq 0\).