### 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 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

*Applications of Mathematics*,

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

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*Applications of Mathematics*, vol. 56, no. 5, pp. 425-457. https://doi.org/10.1007/s10492-011-0025-0

**Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations.** / Milani, Albert; Volkmer, Hans.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

AU - Milani, Albert

AU - Volkmer, Hans

PY - 2011/10/1

Y1 - 2011/10/1

N2 - 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 tu) α iα ju=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 iα jv = h.We then give conditions for the convergence, as t-∞, of the solution of the evolution equation to its stationary state.

AB - 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 tu) α iα ju=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 iα jv = h.We then give conditions for the convergence, as t-∞, of the solution of the evolution equation to its stationary state.

UR - http://www.scopus.com/inward/record.url?scp=84855442145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855442145&partnerID=8YFLogxK

U2 - 10.1007/s10492-011-0025-0

DO - 10.1007/s10492-011-0025-0

M3 - Article

VL - 56

SP - 425

EP - 457

JO - Applications of Mathematics

JF - Applications of Mathematics

SN - 0862-7940

IS - 5

ER -