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 ⊂ Rdis 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→ Rdfor 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.