Title:
|
Periodic solutions to a one-dimensional strongly nonlinear wave equation with strong dissipation (English) |
Author:
|
Herrmann, Leopold |
Language:
|
English |
Journal:
|
Czechoslovak Mathematical Journal |
ISSN:
|
0011-4642 (print) |
ISSN:
|
1572-9141 (online) |
Volume:
|
35 |
Issue:
|
2 |
Year:
|
1985 |
Pages:
|
278-294 |
Summary lang:
|
Russian |
. |
Category:
|
math |
. |
MSC:
|
35B10 |
MSC:
|
35L70 |
idZBL:
|
Zbl 0587.35006 |
idMR:
|
MR787130 |
DOI:
|
10.21136/CMJ.1985.102016 |
. |
Date available:
|
2008-06-09T15:05:00Z |
Last updated:
|
2020-07-28 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/102016 |
. |
Reference:
|
[1] G. Andrews: On the existence of solutions to the equation $u\sb{tt}=u\sb{xxt}+\sigma (u\sb{x})\sb{x}$.J. Differential Eq. 35 (1980), 200-231. MR 0561978 |
Reference:
|
[2] G. Andrews J. M. Ball: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity.J. Differential Eq. 44 (1982), 306-341. MR 0657784, 10.1016/0022-0396(82)90019-5 |
Reference:
|
[3] R. Arima Y. Hasegawa: On global solutions for mixed problem of semi-linear differential equation.Proc. Japan. Acad. 39 (1963), 721-725. MR 0161046 |
Reference:
|
[4] T. K. Caughey J. Ellison: Existence, uniqueness and stability of solutions of a class of nonlinear partial differential equations.J. Math. Anal. Appl. 51 (1975), 1 - 32. MR 0387801, 10.1016/0022-247X(75)90136-5 |
Reference:
|
[5] J. С. Clements: Existence theorems for a quasilinear evolution equation.SIAM J. Appl. Math. 26 (1974), 745-752. Zbl 0252.35044, MR 0372426, 10.1137/0126066 |
Reference:
|
[6] J. С. Clements: On the existence and uniqueness of solutions of the equation $u\sb{tt}-\partial \sigma \sb{i}(u\sb{x\sb{i}})/\partial x\sb{i}-D\sb{N}u\sb{t}=f$.Canad. Math. Bull. 18 (1975), 181-187. MR 0397200 |
Reference:
|
[7] С. M. Dafermos: The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity.J. Differential Eq. 6 (1969), 71 - 86. Zbl 0218.73054, MR 0241831, 10.1016/0022-0396(69)90118-1 |
Reference:
|
[8] P. L. Davis: A quasilinear hyperbolic and related third-order equations.J. Math. Anal. Appl. 51 (1975), 596-606. Zbl 0312.35018, MR 0390514, 10.1016/0022-247X(75)90110-9 |
Reference:
|
[9] Y. Ebihara: On some nonlinear evolution equations with the strong dissipation.J. Differential Eq. 30 (1978), 149-164, II, ibid. 34 (1979), 339-352. MR 0513267, 10.1016/0022-0396(78)90011-6 |
Reference:
|
[10] Y. Ebihara: Some evolution equations with the quasi-linear strong dissipation.J. Math. Pures et Appl. 58 (1979), 229-245. Zbl 0405.35049, MR 0539221 |
Reference:
|
[11] Y. Ebihara: Some evolution equations with linear and quasi-linear strong dissipation.J. Gen. Res. Inst. Fukuoka Univ. 66 (1983), 7-19. Zbl 0536.35052, MR 0730317 |
Reference:
|
[12] W. M. Ewing W. S. Jardetzky F. Press: Elastic waves in layered media.McGraw-Hill series in the geological sciences, New York-Toronto-London 1957. MR 0094967 |
Reference:
|
[13] W. E. Fitzgibbon: Strongly damped quasilinear evolution equations.J. Math. Anal. Appl. 79 (1981), 536-550. Zbl 0476.35040, MR 0606499, 10.1016/0022-247X(81)90043-3 |
Reference:
|
[14] J. M. Greenberg: On the existence, uniqueness, and stability of solutions of the equation $\rho \sb{0}{\germ X}\sb{tt}=E({\germ X}\sb{x}) {\germ X}\sb{xx}+\lambda {\germ X}\sb{xxt}$.J. Math. Anal. Appl. 25 (1969), 575-591. MR 0240473 |
Reference:
|
[15] J. M. Greenberg R. C. MacCamy: On the exponential stability of solutions of $E(u\sb{x})u\sb{xx}+\lambda u\sb{xtx}=\rho u\sb{tt}$.J. Math. Anal. Appl. 31 (1970), 406-417. MR 0273178 |
Reference:
|
[16] J. M. Greenberg R. C. MacCamy V. S. Mizel: On the existence, uniqueness, and stability of solutions of the equation $\sigma \sp{\prime} \,(u\sb{x})u\sb{xx}+\lambda u\sb{xtx}=\rho \sb{0}u\sb{tt}$.J. Math. Mech. 17 (1968), 707-728. MR 0225026 |
Reference:
|
[17] A. Haraux: Nonlinear evolution equations - global behavior of solutions.Lecture Notes in Mathematics Vol. 841, Springer-Verlag, Berlin- Heidelberg-New York 1981. Zbl 0461.35002, MR 0610796, 10.1007/BFb0089606 |
Reference:
|
[18] L. Herrmann: Periodic solutions of a strongly nonlinear wave equation with internal friction.(Czech.) Thesis, Prague 1977, 30 pp. |
Reference:
|
[19] L. Herrmann: Periodic solutions of abstract differential equations: the Fourier method.Czechoslovak Math. J. 30 (1980), 177-206. Zbl 0445.35013, MR 0566046 |
Reference:
|
[20] T. Kakita: Time-periodic solutions of some non-linear evolution equations.Publ. Res. Inst. Math. Sci 9 (1973/74), 477-492. MR 0342868, 10.2977/prims/1195192568 |
Reference:
|
[21] H. Kolsky: Stress waves in Solids.Clarendon Press, Oxford 1953. Zbl 0052.42502 |
Reference:
|
[22] A. I. Kozhanov: An initial-boundary value problem for a class of equations of non-classical type.(Russian.) Differenciaľnye Uravn. 15 (1979), 272-280. (English trans., in: Differential Equations 15 (1979), 186-191.) |
Reference:
|
[23] A. I. Kozhanov N. A. Lar'kin N. N. Janenko: On a regularization of equations of variable type.(Russian.) Dokl. Akad. Nauk SSSR 252 (1980), 525-527. (English trans., in: Soviet Math. Dokl. 21 (1980), 758-761.) MR 0577831 |
Reference:
|
[24] P. A. Lagerstrom J. D. Cole L. Trilling: Problems in the theory of viscous compressible fluids.California Institute of Technology 1949. MR 0041617 |
Reference:
|
[25] J.-L. Lions: Equations différentielles opérationnelles et problèmes aux limites.Springer-Verlag, Berlin-Göttingen-Heidelberg 1961. Zbl 0098.31101, MR 0153974 |
Reference:
|
[26] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod, Paris 1969. Zbl 0189.40603, MR 0259693 |
Reference:
|
[27] J.-L. Lions E. Magenes: Problèrnes aux limites non homogènes et applications I, III.Dunod, Paris 1968. |
Reference:
|
[28] V. Lovicar: Theorem of Fréchet and asymptotically almost periodic solutions of some nonlinear equations of hyperbolic type.Nonlinear evolution equations and potential theory. Proc. of a Summer School held in September 1973 at Podhradí near Ledeč on Sázava. Ed. Josef Král. Academia, Prague 1975. MR 0481401 |
Reference:
|
[29] R. C. MacCamy V. C. Mizel: Existence and non-existence in the large of solutions of quasilinear wave equations.Arch. Rational Mech. Anal. 25 (1967), 299-320. MR 0216165, 10.1007/BF00250932 |
Reference:
|
[30] E. J. McShane: Integration.Princeton University Press 1947. Zbl 0033.05302, MR 0082536 |
Reference:
|
[31] H. Pecher: On global regular solutions of third order partial differential equations.J. Math. Anal. Appl. 73 (1980), 278-299. Zbl 0429.35057, MR 0560948, 10.1016/0022-247X(80)90033-5 |
Reference:
|
[32] M. A. Prestel: Forced oscillations for the solutions of a nonlinear hyperbolic equation.Nonlinear Anal. 6 (1982), 209-216. Zbl 0504.35065, MR 0654313, 10.1016/0362-546X(82)90089-X |
Reference:
|
[33] C. O. A. Sowunmi: On the existence of periodic solutions of the equation $\rho \sb{tt}u-(\sigma (u\sb{x}))\sb{x}-\lambda u\sb{xtx}-f=0$.Rend. Ist. Mat. Univ. Trieste 8 (1976), 58-68. MR 0430486 |
Reference:
|
[34] I. Straškraba O. Vejvoda: Periodic solutions to abstract differential equations.Czechoslovak Math. J. 23 (1973), 635-669, 27 (1977), 511-513. MR 0499577 |
Reference:
|
[35] A. E. Taylor: Introduction to functional analysis.J. Wiley and Sons, Inc., New York 1958. Zbl 0081.10202, MR 0098966 |
Reference:
|
[36] M. Tsutsumi: Some nonlinear evolution equations of second order.Proc. Japan Acad. 47 (1971), 950-955. Zbl 0258.35017, MR 0312023 |
Reference:
|
[37] G. F. Webb: Existence and asymptotic behavior for a strongly damped nonlinear wave equation.Canad. J. Math. 32 (1980), 631-643. Zbl 0414.35046, MR 0586981, 10.4153/CJM-1980-049-5 |
Reference:
|
[38] Y. Yamada: Note on certain nonlinear evolution equations of second order.Proc. Japan Acad. 55 (1979), 167-171. Zbl 0436.47054, MR 0533540 |
Reference:
|
[39] Y. Yamada: Some remarks on the equation $y\sb{tt}-\sigma (y\sb{x})$ $y\sb{xx}-y\sb{xtx}=f$.Osaka J. Math. 17 (1980), 303-323. MR 0587752 |
Reference:
|
[40] Y. Yamada: Quasilinear wave equations and related nonlinear evolution equations.Nagoya Math. J. 84(1981), 31-83. Zbl 0472.35052, MR 0641147, 10.1017/S0027763000019553 |
. |