Blow-up of the energy at infinity for 2 by 2 systems

The goal of this paper is to study the long time behavior of the energy of solutions to 2 by 2 linear hyperbolic systems. Some blow-up results at t=∞ are given for a large class of initial data of the Cauchy problem. We shall prove the optimality of these results for a special class of systems. We present also an example of generalized energy conservation under a Cm stabilization condition.