Linear Hyperbolic Systems in Domains with Growing Cracks

Abstract

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.