| Title:
|
Global in time solutions to quasilinear telegraph equations involving operators with time delay (English) |
| Author:
|
Feireisl, Eduard |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
36 |
| Issue:
|
6 |
| Year:
|
1991 |
| Pages:
|
456-468 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The existence of small global (in time) solutions to an abstract evolution equation containing a damping term is proved. The result is then applied to fully nonlinear telegraph equations and to nonlinear equations involving operators with time delay. (English) |
| Keyword:
|
quasilinear telegraph equations |
| Keyword:
|
bounded solutions |
| Keyword:
|
time-periodic solutions |
| Keyword:
|
time delay |
| Keyword:
|
small global solution |
| Keyword:
|
abstract evolution equation |
| Keyword:
|
nonlinear coefficients |
| Keyword:
|
nonlinear right-hand side |
| MSC:
|
35A05 |
| MSC:
|
35B35 |
| MSC:
|
35L70 |
| MSC:
|
45G10 |
| MSC:
|
45K05 |
| idZBL:
|
Zbl 0752.45012 |
| idMR:
|
MR1134922 |
| DOI:
|
10.21136/AM.1991.104482 |
| . |
| Date available:
|
2008-05-20T18:42:41Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104482 |
| . |
| Reference:
|
[1] A. B. Aliev: The global solvability of unilateral problems for quasilinear operators of the hyperbolic type (Russian).Dokl. Akad. Nauk SSSR 298 (5) (1988), 1033-1036. MR 0939671 |
| Reference:
|
[2] A. Arosio: Global (in time) solution of the approximate non-linear string equation of G. F. Carrier and R. Narasimha.Comment. Math. Univ. Carolinae 26 (1) (1985), 169-172. MR 0797299 |
| Reference:
|
[3] A. Arosio: Linear second order differential equations in Hilbert spaces - the Cauchy problem and asymptotic behaviour for large time:.Arch. Rational Mech. Anal. 86 (2) (1984), 147-180. Zbl 0563.35041, MR 0751306, 10.1007/BF00275732 |
| Reference:
|
[4] P. Biler: Remark on the decay for damped string and beam equations.Nonlinear Anal. 10 (9) (1984), 839-842. MR 0856867, 10.1016/0362-546X(86)90071-4 |
| Reference:
|
[5] E. Feireisl: Small time-periodic solutions to a nonlinear equation of a vibrating string.Apl. mat. 32 (6) (1987), 480-490. Zbl 0653.35063, MR 0916063 |
| Reference:
|
[6] Z. Kamont J. Turo: On the Cauchy problem for quasilinear hyperbolic systems with a retarded argument.Ann. Mat. Рurа Appl. 143 (4) (1986), 235 - 246. MR 0859605, 10.1007/BF01769218 |
| Reference:
|
[7] T. Kato: Quasilinear equations of evolution with applications to partial differential equations.Lectures Notes in Mathematics 448, 25 - 70. Springer, Berlin 1975. MR 0407477, 10.1007/BFb0067080 |
| Reference:
|
[8] P. Krejčí: Hard implicit function theorem and small periodic solutions to partial differential equations.Comment. Math. Univ. Carolinae 25 (1984), 519-536. MR 0775567 |
| Reference:
|
[9] J. L. Lions E. Magenes: Problèmes aux limites non homogènes et applications I.Dunod, Paris 1968. |
| Reference:
|
[10] A. Matsumura: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation.Publ. RIMS Kyoto Univ. 13 (1977), 349-379. Zbl 0371.35030, MR 0470507, 10.2977/prims/1195189813 |
| Reference:
|
[11] L. A. Medeiros: On a new class of nonlinear wave equations.J. Math. Anal. Appl. 69 (1) (1979), 252-262. Zbl 0407.35051, MR 0535295, 10.1016/0022-247X(79)90192-6 |
| Reference:
|
[12] H. Petzeltová M. Štědrý: Time periodic solutions of telegraph equations in n spatial variables.Časopis Pěst. Mat. 109 (1984), 60-73. MR 0741209 |
| Reference:
|
[13] H. Poorkarimi J. Wiener: Bounded solutions of nonlinear hyperbolic equations with delay.Lecture Notes in Pure and Appl. Math. 109 (Dekker), 1987. MR 0912327 |
| Reference:
|
[14] P. H. Rabmowitz: Periodic solutions of nonlinear hyperbolic partial differential equations II.Comm. Pure Appl. Math. 22 (1969), 15-39. 10.1002/cpa.3160220103 |
| Reference:
|
[15] Y. Shibata Y. Tsutsumi: Local existence of solution for the initial boundary value problem of fully nonlinear wave equation.Nonlinear Anal. 11 (3) (1987), 335-365. MR 0881723, 10.1016/0362-546X(87)90051-4 |
| Reference:
|
[16] M. Štědrý: Small time-periodic solutions to fully nonlinear telegraph equations in more spatial dimensions.preprint. Ann. Inst. Henri Poincaré 6 (3) (1989), 209-232. MR 0995505 |
| Reference:
|
[17] Vejvoda O., al.: Partial differential equations: Time periodic solutions.Martinus Nijhoff Publ., 1982. Zbl 0501.35001 |
| . |
