| Title:
|
Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations (English) |
| Author:
|
Milani, Albert |
| Author:
|
Volkmer, Hans |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
56 |
| Issue:
|
5 |
| Year:
|
2011 |
| Pages:
|
425-457 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasilinear evolution equation |
| Keyword:
|
quasilinear elliptic equation |
| Keyword:
|
a priori estimates |
| Keyword:
|
global existence |
| Keyword:
|
asymptotic behavior |
| Keyword:
|
stationary solutions |
| MSC:
|
35A01 |
| MSC:
|
35B35 |
| MSC:
|
35B40 |
| MSC:
|
35J15 |
| MSC:
|
35J60 |
| MSC:
|
35L15 |
| MSC:
|
35L70 |
| idZBL:
|
Zbl 1249.35072 |
| idMR:
|
MR2852065 |
| DOI:
|
10.1007/s10492-011-0025-0 |
| . |
| Date available:
|
2011-09-22T14:17:12Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141617 |
| . |
| Reference:
|
[1] Adams, R., Fournier, J.: Sobolev Spaces, 2nd ed.Academic Press New York (2003). Zbl 1098.46001, MR 2424078 |
| Reference:
|
[2] Cattaneo, C.: Sulla Conduzione del Calore.Atti Semin. Mat. Fis., Univ. Modena 3 (1949), 83-101 Italian. Zbl 0035.26203, MR 0032898 |
| Reference:
|
[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). |
| Reference:
|
[4] Haus, H. A.: Waves and Fields in Optoelectronics.Prentice Hall (1984). |
| Reference:
|
[5] Kato, T.: Abstract Differential Equations and Nonlinear Mixed Problems. Fermian Lectures.Academie Nazionale dei Licei Pisa (1985). MR 0930267 |
| Reference:
|
[6] Klainerman, S.: Global existence for nonlinear wave equations.Commun. Pure Appl. Math. 33 (1980), 43-101. Zbl 0405.35056, MR 0544044, 10.1002/cpa.3160330104 |
| Reference:
|
[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). MR 1654353 |
| Reference:
|
[8] Matsumura, A.: On the asymptotic behavior of solutions to semi-linear wave equations.Publ. Res. Inst. Mat. Sci., Kyoto Univ. 12 (1976), 169-189. MR 0420031, 10.2977/prims/1195190962 |
| Reference:
|
[9] Matsumura, A.: Global existence and asymptotics of the solutions of second-order quasilinear hyperbolic equations with first-order dissipation.Publ. Res. Inst. Mat. Sci., Kyoto Univ. 13 (1977), 349-379. MR 0470507, 10.2977/prims/1195189813 |
| Reference:
|
[10] Milani, A.: The quasi-stationary Maxwell equations as singular limit of the complete equations: The quasi-linear case.J. Math. Anal. Appl. 102 (1984), 251-274. Zbl 0551.35006, MR 0751358, 10.1016/0022-247X(84)90218-X |
| Reference:
|
[11] Milani, A.: Global existence via singular perturbations for quasilinear evolution equations.Adv. Math. Sci. Appl. 6 (1996), 419-444. Zbl 0868.35008, MR 1411976 |
| Reference:
|
[12] Mizohata, S.: The Theory of Partial Differential Equations.Cambridge University Press London (1973). Zbl 0263.35001, MR 0599580 |
| Reference:
|
[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. Zbl 0174.47801, MR 0199523 |
| Reference:
|
[14] Ponce, G.: Global existence of small solutions to a class of nonlinear evolution equations.Nonlin. Anal., Theory Methods Appl. 9 (1985), 399-418. Zbl 0576.35023, MR 0785713, 10.1016/0362-546X(85)90001-X |
| Reference:
|
[15] Racke, R.: Non-homogeneous nonlinear damped wave equations in unbounded domains.Math. Methods Appl. Sci. 13 (1990), 481-491. Zbl 0728.35071, 10.1002/mma.1670130604 |
| Reference:
|
[16] Racke, R.: Lectures on Nonlinear Evolution Equations. Initial Value Problems.Vieweg Braunschweig (1992). Zbl 0811.35002, MR 1158463 |
| Reference:
|
[17] Schochet, S.: The instant-response limit in Whitham's nonlinear traffic flow model: Uniform well-posedness and global existence.Asymptotic Anal. 1 (1988), 263-282. Zbl 0685.35099, MR 0972301, 10.3233/ASY-1988-1401 |
| Reference:
|
[18] Yang, H., Milani, A.: On the diffusion phenomenon of quasilinear hyperbolic waves.Bull. Sci. Math. 124 (2000), 415-433. Zbl 0959.35126, MR 1781556, 10.1016/S0007-4497(00)00141-X |
| . |
