Title | Linear Hyperbolic Systems in Domains with Growing Cracks |

Publication Type | Journal Article |

Year of Publication | 2017 |

Authors | Caponi, M |

Volume | 85 |

Issue | 1 |

Pagination | 149 - 185 |

Date Published | 2017/06/01 |

ISBN Number | 1424-9294 |

Abstract | 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. |

URL | https://doi.org/10.1007/s00032-017-0268-7 |

Short Title | Milan Journal of Mathematics |

## Linear Hyperbolic Systems in Domains with Growing Cracks

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