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

Albert J. Milani; Hans Volkmer

Applications of Mathematics (2011)

  • Volume: 56, Issue: 5, page 425-457
  • ISSN: 0862-7940

Abstract

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We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $$ u_{tt} + 2 u_t - a_{ij}(u_t,\nabla u)\partial _i\partial _j u = f $$ 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 $$ -a_{ij}(0,\nabla v)\partial _i\partial _j v=h. $$ We then give conditions for the convergence, as $t\to \infty $, of the solution of the evolution equation to its stationary state.

How to cite

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Milani, Albert J., and Volkmer, Hans. "Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations." Applications of Mathematics 56.5 (2011): 425-457. <http://eudml.org/doc/196470>.

@article{Milani2011,
abstract = {We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $$ u\_\{tt\} + 2 u\_t - a\_\{ij\}(u\_t,\nabla u)\partial \_i\partial \_j u = f $$ 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 $$ -a\_\{ij\}(0,\nabla v)\partial \_i\partial \_j v=h. $$ We then give conditions for the convergence, as $t\to \infty $, of the solution of the evolution equation to its stationary state.},
author = {Milani, Albert J., Volkmer, Hans},
journal = {Applications of Mathematics},
keywords = {quasilinear evolution equation; quasilinear elliptic equation; global existence; asymptotic behavior; stationary solution},
language = {eng},
number = {5},
pages = {425-457},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations},
url = {http://eudml.org/doc/196470},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Milani, Albert J.
AU - Volkmer, Hans
TI - Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 5
SP - 425
EP - 457
AB - We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $$ u_{tt} + 2 u_t - a_{ij}(u_t,\nabla u)\partial _i\partial _j u = f $$ 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 $$ -a_{ij}(0,\nabla v)\partial _i\partial _j v=h. $$ We then give conditions for the convergence, as $t\to \infty $, of the solution of the evolution equation to its stationary state.
LA - eng
KW - quasilinear evolution equation; quasilinear elliptic equation; global existence; asymptotic behavior; stationary solution
UR - http://eudml.org/doc/196470
ER -

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