Linear Hyperbolic Systems in Domains with Growing Cracks

We consider the hyperbolic system ü - div(A∇u) = f in the time varying cracked domain Ω \ Γt, where the set Ω ⊂ Rd is open, bounded, and with Lipschitz boundary, the cracks Γt, t ∈ [0,T], are closed subsets of Ω ¯, increasing with respect to inclusion, and u(t) : Ω \ Γt→ Rd for every t ∈ [0,T]. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈ - div (B∇v) + a∇v - 2∇v˙b = g on the fixed domain Ω \ Γ0. 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.