Linear Hyperbolic Systems in Domains with Growing Cracks

Abstract

We consider the hyperbolic system u\({ - {\rm div} (\mathbb{A} \nabla u) = f}\) 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 v\({ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}\) 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.