The wave equation on domains with cracks growing on a prescribed path: Existence, uniqueness, and continuous dependence on the data

Abstract

Given a bounded open set W Ω⊂d with Lipschitz boundary and an increasing family Gt, t ∈ [0, T], of closed subsets of Ω, we analyze the scalar wave equation ü - div(A∇u) = f in the time varying cracked domains Ω\Gt. Here we assume that the sets Gt are contained into a prescribed (d - 1)-manifold of class C2. Our approach relies on a change of variables: Recasting the problem on the reference configuration Ω\G0, we are led to consider a hyperbolic problem of the form v - div(B∇v) + a ∇v- 2b ∇v = g in Ω\G0. Under suitable assumptions on the regularity of the change of variables that transforms Ω\Γt into Ω\Γ0, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.