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 t t + 2 u t - a i j ( u t , ∇ u ) ∂ i ∂ 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 i j ( 0 , ∇ v ) ∂ i ∂ j v = h . We then give conditions for the convergence, as t → ∞ , 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\rightarrow \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; a priori estimates; global existence; asymptotic behavior; stationary solutions; 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\rightarrow \infty $, of the solution of the evolution equation to its stationary state.
LA - eng
KW - quasilinear evolution equation; quasilinear elliptic equation; a priori estimates; global existence; asymptotic behavior; stationary solutions; quasilinear evolution equation; quasilinear elliptic equation; global existence; asymptotic behavior; stationary solution
UR - http://eudml.org/doc/196470
ER -

References

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  1. Adams, R., Fournier, J., Sobolev Spaces, 2nd ed, Academic Press New York (2003). (2003) Zbl1098.46001MR2424078
  2. Cattaneo, C., Sulla Conduzione del Calore, Atti Semin. Mat. Fis., Univ. Modena 3 (1949), 83-101 Italian. (1949) Zbl0035.26203MR0032898
  3. Hadeler, K P., Random walk systems and reaction telegraph equations, In: Dynamical Systems and their Applications S. van Strien, S. V. Lunel Royal Academy of the Netherlands (1995). (1995) 
  4. Haus, H. A., Waves and Fields in Optoelectronics, Prentice Hall (1984). (1984) 
  5. Kato, T., Abstract Differential Equations and Nonlinear Mixed Problems. Fermian Lectures, Academie Nazionale dei Licei Pisa (1985). (1985) MR0930267
  6. Klainerman, S., 10.1002/cpa.3160330104, Commun. Pure Appl. Math. 33 (1980), 43-101. (1980) Zbl0405.35056MR0544044DOI10.1002/cpa.3160330104
  7. Li, Ta-Tsien, Nonlinear Heat Conduction with Finite Speed of Popagation. Proceedings of the China-Japan Symposium on Reaction Diffusion Equations and their Applcations to Computational Aspects, World Scientific Singapore (1997). (1997) MR1654353
  8. Matsumura, A., 10.2977/prims/1195190962, Publ. Res. Inst. Mat. Sci., Kyoto Univ. 12 (1976), 169-189. (1976) MR0420031DOI10.2977/prims/1195190962
  9. Matsumura, A., 10.2977/prims/1195189813, Publ. Res. Inst. Mat. Sci., Kyoto Univ. 13 (1977), 349-379. (1977) MR0470507DOI10.2977/prims/1195189813
  10. Milani, A., 10.1016/0022-247X(84)90218-X, J. Math. Anal. Appl. 102 (1984), 251-274. (1984) Zbl0551.35006MR0751358DOI10.1016/0022-247X(84)90218-X
  11. Milani, A., Global existence via singular perturbations for quasilinear evolution equations, Adv. Math. Sci. Appl. 6 (1996), 419-444. (1996) Zbl0868.35008MR1411976
  12. Mizohata, S., The Theory of Partial Differential Equations, Cambridge University Press London (1973). (1973) Zbl0263.35001MR0599580
  13. Moser, J., A rapidly convergent iteration method and non-linear differential equations, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 20 (1966), 265-315. (1966) Zbl0174.47801MR0199523
  14. Ponce, G., 10.1016/0362-546X(85)90001-X, Nonlin. Anal., Theory Methods Appl. 9 (1985), 399-418. (1985) Zbl0576.35023MR0785713DOI10.1016/0362-546X(85)90001-X
  15. Racke, R., 10.1002/mma.1670130604, Math. Methods Appl. Sci. 13 (1990), 481-491. (1990) Zbl0728.35071DOI10.1002/mma.1670130604
  16. Racke, R., Lectures on Nonlinear Evolution Equations. Initial Value Problems, Vieweg Braunschweig (1992). (1992) Zbl0811.35002MR1158463
  17. Schochet, S., 10.3233/ASY-1988-1401, Asymptotic Anal. 1 (1988), 263-282. (1988) Zbl0685.35099MR0972301DOI10.3233/ASY-1988-1401
  18. Yang, H., Milani, A., 10.1016/S0007-4497(00)00141-X, Bull. Sci. Math. 124 (2000), 415-433. (2000) Zbl0959.35126MR1781556DOI10.1016/S0007-4497(00)00141-X

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