### Abstract

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u _{t} _{t}+2u _{t}-a _{ij}(u _{t}u) α _{i}α _{j}u=f corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation -a _{ij}(0,v)α _{i}α _{j}v = h.We then give conditions for the convergence, as t-∞, of the solution of the evolution equation to its stationary state.

Original language | English |
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Pages (from-to) | 425-457 |

Number of pages | 33 |

Journal | Applications of Mathematics |

Volume | 56 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 1 2011 |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

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## Cite this

Milani, A., & Volkmer, H. (2011). Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations.

*Applications of Mathematics*,*56*(5), 425-457. https://doi.org/10.1007/s10492-011-0025-0