Asymptotic behaviour for symmetric hyperbolic dissipative systems
The existence, uniqueness and asymptotic behaviour of solutions to the system \[ E{\partial \over \partial t}U={\partial \over \partial x_i}(A^i U)+BU+f \] is investigated in a bounded domain \(\Omega \subset \mathbb{R}^3\). \(E\) and \(A^i\) are symmetric \(m\times m\) matrices, \(E\) is positive definite and \[ \sup \Biggl\{ -{A^i(x)\nu _i\xi \cdot \xi \over E(x)\xi \cdot \xi }:\;x\in \Omega ,\;\xi \in \mathbb{R}^m, \sum _{i=1}^3 \nu _i=1\Biggr\}<\infty . \] A portion of of the boundary of \(\Omega \) is required to be an energy absorbing surface, so that a damping mechanism is achieved.
scientific article; zbMATH DE number 1153406