Linear Hyperbolic Systems in Domains with Growing Cracks

We consider the hyperbolic system u in the time varying cracked domain , where the set is open, bounded, and with Lipschitz boundary, the cracks , are closed subsets of , increasing with respect to inclusion, and for every . We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system vI on the fixed domain . Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.