*Published Paper*

**Inserted:** 27 jan 2017

**Last Updated:** 27 may 2017

**Journal:** Milan J. Math.

**Volume:** 85

**Pages:** 149-185

**Year:** 2017

**Doi:** 10.1007/s00032-017-0268-7

**Links:**
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**Abstract:**

We consider the hyperbolic system $\ddot u-{\rm div}\,(\mathbb A\nabla u)=f$ in the time varying cracked domain $\Omega\setminus\Gamma_t$, 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 $\overline\Omega$, increasing with respect to inclusion, and $u(t):\Omega\setminus\Gamma_t\to\mathbb R^d$ for every $t\in[0,T]$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system $\ddot v-{\rm div}\,(\mathbb B\nabla v)+\mathbf a\nabla v -2\nabla\dot vb=g$ on the fixed domain $\Omega\setminus\Gamma_0$. 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.

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