Linear Hyperbolic Systems in Domains with Growing Cracks.

We consider the hyperbolic system ü $${ - {\rm div} (\mathbb{A} \nabla u) = f}$$ in the time varying cracked domain $${\Omega \backslash \Gamma_t}$$ , where the set $${\Omega \subset \mathbb{R}^d}$$ is open, bounded, and with Lipschitz boundary, the cracks $${\Gamma_t, t \in [0, T]}$$ , are closed subsets of $${\bar{\Omega}}$$ , increasing with respect to inclusion, and $${u(t) : \Omega \backslash \Gamma_t \rightarrow \mathbb{R}^d}$$ for every $${t \in [0, T]}$$ . We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈ $${ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}$$ on the fixed domain $${\Omega \backslash \Gamma_0}$$ . 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. [ABSTRACT FROM AUTHOR]