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.

中文翻译：

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裂纹扩展域中的线性双曲系统

我们考虑时变裂变域\（{\ Omega \反斜杠\ Gamma_t} \）中的双曲系统ü \（{-{\ rm div}（\ mathbb {A} \ nabla u）= f } \），其中集合\（{\ Omega \ subset \ mathbb {R} ^ d} \）是开放的，有界的，并且具有Lipschitz边界，裂缝\（{\ Gamma_t，t \ in [0，T]} \）是的封闭子集\（{\巴{\欧米茄}} \） ，相对于增加至夹杂物，和\（{U（t）的：\欧米茄\反斜杠\ Gamma_t \ RIGHTARROW \ mathbb {R} ^ d} \）为每个\（{t \ in [0，T]} \）。我们假设变量存在适当的正则变化，这将我们的问题简化为变换后的系统v̈ \（{-{\ rm div}（\ mathbb {B} \ nabla v）+ a \ nabla v-2 \ nabla \ dot {v} b = g} \）在固定域\（{\ Omega \反斜杠\ Gamma_0} \）。在这些假设下，我们获得了这两个问题的弱解的存在性和唯一性。此外，我们展示了函数v的能量相等，这使我们能够证明两个系统的连续依赖结果。在标量情况下，已经在[3，7]中进行了相同的研究。