Periodic solutions to a one-dimensional strongly nonlinear wave equation with strong dissipation

Leopold Herrmann

Czechoslovak Mathematical Journal (1985)

  • Volume: 35, Issue: 2, page 278-294
  • ISSN: 0011-4642

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Herrmann, Leopold. "Periodic solutions to a one-dimensional strongly nonlinear wave equation with strong dissipation." Czechoslovak Mathematical Journal 35.2 (1985): 278-294. <http://eudml.org/doc/13511>.

@article{Herrmann1985,
author = {Herrmann, Leopold},
journal = {Czechoslovak Mathematical Journal},
keywords = {Time-periodic solutions; wave propagation; diffusion; exponential decay; semigroup; existence of weak -periodic solutions},
language = {eng},
number = {2},
pages = {278-294},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Periodic solutions to a one-dimensional strongly nonlinear wave equation with strong dissipation},
url = {http://eudml.org/doc/13511},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Herrmann, Leopold
TI - Periodic solutions to a one-dimensional strongly nonlinear wave equation with strong dissipation
JO - Czechoslovak Mathematical Journal
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 2
SP - 278
EP - 294
LA - eng
KW - Time-periodic solutions; wave propagation; diffusion; exponential decay; semigroup; existence of weak -periodic solutions
UR - http://eudml.org/doc/13511
ER -

References

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  1. G. Andrews, On the existence of solutions to the equation u t t = u x x t + σ ( u x ) x , J. Differential Eq. 35 (1980), 200-231. (1980) MR0561978
  2. G. Andrews J. M. Ball, 10.1016/0022-0396(82)90019-5, J. Differential Eq. 44 (1982), 306-341. (1982) MR0657784DOI10.1016/0022-0396(82)90019-5
  3. R. Arima Y. Hasegawa, On global solutions for mixed problem of semi-linear differential equation, Proc. Japan. Acad. 39 (1963), 721-725. (1963) MR0161046
  4. T. K. Caughey J. Ellison, 10.1016/0022-247X(75)90136-5, J. Math. Anal. Appl. 51 (1975), 1 - 32. (1975) MR0387801DOI10.1016/0022-247X(75)90136-5
  5. J. С. Clements, 10.1137/0126066, SIAM J. Appl. Math. 26 (1974), 745-752. (1974) Zbl0252.35044MR0372426DOI10.1137/0126066
  6. J. С. Clements, On the existence and uniqueness of solutions of the equation u t t - σ i ( u x i ) / x i - D N u t = f , Canad. Math. Bull. 18 (1975), 181-187. (1975) MR0397200
  7. С. M. Dafermos, 10.1016/0022-0396(69)90118-1, J. Differential Eq. 6 (1969), 71 - 86. (1969) Zbl0218.73054MR0241831DOI10.1016/0022-0396(69)90118-1
  8. P. L. Davis, 10.1016/0022-247X(75)90110-9, J. Math. Anal. Appl. 51 (1975), 596-606. (1975) Zbl0312.35018MR0390514DOI10.1016/0022-247X(75)90110-9
  9. Y. Ebihara, 10.1016/0022-0396(78)90011-6, J. Differential Eq. 30 (1978), 149-164, II, ibid. 34 (1979), 339-352. (1978) MR0513267DOI10.1016/0022-0396(78)90011-6
  10. Y. Ebihara, Some evolution equations with the quasi-linear strong dissipation, J. Math. Pures et Appl. 58 (1979), 229-245. (1979) Zbl0405.35049MR0539221
  11. Y. Ebihara, Some evolution equations with linear and quasi-linear strong dissipation, J. Gen. Res. Inst. Fukuoka Univ. 66 (1983), 7-19. (1983) Zbl0536.35052MR0730317
  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. (1957) MR0094967
  13. W. E. Fitzgibbon, 10.1016/0022-247X(81)90043-3, J. Math. Anal. Appl. 79 (1981), 536-550. (1981) Zbl0476.35040MR0606499DOI10.1016/0022-247X(81)90043-3
  14. J. M. Greenberg, On the existence, uniqueness, and stability of solutions of the equation ρ 0 𝔛 t t = E ( 𝔛 x ) 𝔛 x x + λ 𝔛 x x t , J. Math. Anal. Appl. 25 (1969), 575-591. (1969) MR0240473
  15. J. M. Greenberg R. C. MacCamy, On the exponential stability of solutions of E ( u x ) u x x + λ u x t x = ρ u t t , J. Math. Anal. Appl. 31 (1970), 406-417. (1970) MR0273178
  16. J. M. Greenberg R. C. MacCamy V. S. Mizel, On the existence, uniqueness, and stability of solutions of the equation σ ' ( u x ) u x x + λ u x t x = ρ 0 u t t , J. Math. Mech. 17 (1968), 707-728. (1968) MR0225026
  17. A. Haraux, 10.1007/BFb0089606, Lecture Notes in Mathematics Vol. 841, Springer-Verlag, Berlin- Heidelberg-New York 1981. (1981) Zbl0461.35002MR0610796DOI10.1007/BFb0089606
  18. L. Herrmann, Periodic solutions of a strongly nonlinear wave equation with internal friction, (Czech.) Thesis, Prague 1977, 30 pp. (1977) 
  19. L. Herrmann, Periodic solutions of abstract differential equations: the Fourier method, Czechoslovak Math. J. 30 (1980), 177-206. (1980) Zbl0445.35013MR0566046
  20. T. Kakita, 10.2977/prims/1195192568, Publ. Res. Inst. Math. Sci 9 (1973/74), 477-492. (1973) MR0342868DOI10.2977/prims/1195192568
  21. H. Kolsky, Stress waves in Solids, Clarendon Press, Oxford 1953. (1953) Zbl0052.42502
  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.) (1979) 
  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.) (1980) MR0577831
  24. P. A. Lagerstrom J. D. Cole L. Trilling, Problems in the theory of viscous compressible fluids, California Institute of Technology 1949. (1949) MR0041617
  25. J.-L. Lions, Equations différentielles opérationnelles et problèmes aux limites, Springer-Verlag, Berlin-Göttingen-Heidelberg 1961. (1961) Zbl0098.31101MR0153974
  26. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris 1969. (1969) Zbl0189.40603MR0259693
  27. J.-L. Lions E. Magenes, Problèrnes aux limites non homogènes et applications I, III, Dunod, Paris 1968. (1968) 
  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. (1973) MR0481401
  29. R. C. MacCamy V. C. Mizel, 10.1007/BF00250932, Arch. Rational Mech. Anal. 25 (1967), 299-320. (1967) MR0216165DOI10.1007/BF00250932
  30. E. J. McShane, Integration, Princeton University Press 1947. (1947) Zbl0033.05302MR0082536
  31. H. Pecher, 10.1016/0022-247X(80)90033-5, J. Math. Anal. Appl. 73 (1980), 278-299. (1980) Zbl0429.35057MR0560948DOI10.1016/0022-247X(80)90033-5
  32. M. A. Prestel, 10.1016/0362-546X(82)90089-X, Nonlinear Anal. 6 (1982), 209-216. (1982) Zbl0504.35065MR0654313DOI10.1016/0362-546X(82)90089-X
  33. C. O. A. Sowunmi, On the existence of periodic solutions of the equation ρ t t u - ( σ ( u x ) ) x - λ u x t x - f = 0 , Rend. Ist. Mat. Univ. Trieste 8 (1976), 58-68. (1976) MR0430486
  34. I. Straškraba O. Vejvoda, Periodic solutions to abstract differential equations, Czechoslovak Math. J. 23 (1973), 635-669, 27 (1977), 511-513. (1973) MR0499577
  35. A. E. Taylor, Introduction to functional analysis, J. Wiley and Sons, Inc., New York 1958. (1958) Zbl0081.10202MR0098966
  36. M. Tsutsumi, Some nonlinear evolution equations of second order, Proc. Japan Acad. 47 (1971), 950-955. (1971) Zbl0258.35017MR0312023
  37. G. F. Webb, 10.4153/CJM-1980-049-5, Canad. J. Math. 32 (1980), 631-643. (1980) Zbl0414.35046MR0586981DOI10.4153/CJM-1980-049-5
  38. Y. Yamada, Note on certain nonlinear evolution equations of second order, Proc. Japan Acad. 55 (1979), 167-171. (1979) Zbl0436.47054MR0533540
  39. Y. Yamada, Some remarks on the equation y t t - σ ( y x ) y x x - y x t x = f , Osaka J. Math. 17 (1980), 303-323. (1980) MR0587752
  40. Y. Yamada, 10.1017/S0027763000019553, Nagoya Math. J. 84(1981), 31-83. (1981) Zbl0472.35052MR0641147DOI10.1017/S0027763000019553

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