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Time-periodic solutions to quasilinear telegraph equations with large data

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References

  1. Amann, H., Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J. Math. Anal. Appl. 65, 432–467 (1978).

    Google Scholar 

  2. Amann, H., Periodic solutions of semi-linear parabolic equations. Nonlinear Analysis: A collection of papers in honor of Erich Rothe, pp. 1–29. New York: Academic Press, 1978.

    Google Scholar 

  3. Bloom, F., On the damped nonlinear evolution equation wtt=σ(w)xx-yw t. J. Math. Anal. Appl. 96, 551–583 (1983).

    Google Scholar 

  4. Cannon, J. R., The one-dimensional heat equation, Reading: Addison-Wesley, 1984.

    Google Scholar 

  5. Chueh, K. N., & Conley, C. C., & Smoller, J. A., Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26, 373–392 (1977).

    Google Scholar 

  6. Craig, W., A bifurcation theory for periodic solutions of nonlinear dissipative hyperbolic equations. Ann. Sci. Norm. Sup. Pisa Ser. IV-Vol. 10, 125–167 (1983).

    Google Scholar 

  7. Dafermos, C. M., Estimates for conservation laws with little viscosity. SIAM J. Math. Anal. 18, 409–421 (1987).

    Google Scholar 

  8. DiPerna, R. J., Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292 (2), 383–420 (1985).

    Google Scholar 

  9. DiPerna, R. J., Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82, 27–70 (1983).

    Google Scholar 

  10. Henry, D., Geometric theory of semilinear parabolic equations. Lecture Notes in Math. 840, Springer-Verlag, 1981.

  11. Krejčí, P., Hard implicit function theorem and small periodic solutions to partial differential equations. Comment. Math. Univ. Carolinae 25, 519–536 (1984).

    Google Scholar 

  12. Kufner, A., John, O., & Fučík, S., Function spaces. Prague: Academia, 1977.

    Google Scholar 

  13. Matsumura, A., Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation. Publ. RIMS Kyoto Univ. 13, 349–379 (1977).

    Google Scholar 

  14. Milani, A. J., Global existence for quasi-linear dissipative wave equations with large data and small parameter. Math. Z. 198, 291–297 (1988).

    Google Scholar 

  15. Milani, A. J., Long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term II. Nonlinear Anal. 11, 1371–1381 (1987).

    Google Scholar 

  16. Milani, A. J., Time periodic smooth solutions of hyperbolic quasilinear equations with dissipation term and their approximation by parabolic equations. Ann. Mat. Pura Appl. 140 (4), 331–344 (1985).

    Google Scholar 

  17. Murat, F., A survey on compensated compactness. Contributions to modern calculus of variations (L. Cesari ed.), pp. 145–183. Pitman Research Notes in Math. Ser. 148, Longman, 1987.

  18. Nishida, T., Nonlinear hyperbolic equations and related topics in fluid dynamics. Publications Mathématiques d'Orsay 78.02, Univ. Paris Sud, 1978.

  19. Petzeltová, H., Application of Moser's method to a certain type of evolution equations. Czechoslovak Math. J. 33, 427–434 (1983).

    Google Scholar 

  20. Petzeltová, H., & Štědrý, M., Time periodic solutions of telegraph equations in n spatial variables. Časopis Pěst. Mat. 109, 60–73 (1984).

    Google Scholar 

  21. Rabinowitz, P. H.. Periodic solutions of nonlinear hyperbolic partial differential equations II. Comm. Pure Appl. Math. 22, 15–39 (1969).

    Google Scholar 

  22. Rascle, M., Un résultat de “compacité par compensation à coefficients variables”. Application à l'élasticité non linéaire. C. R. Acad. Sc. Paris 302 Sér. I 8, 311–314 (1986).

    Google Scholar 

  23. Serre, D., La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations a une dimension d'éspace. J. Math. Pures Appl. 65, 423–468 (1986).

    Google Scholar 

  24. Slemrod, M.. Damped conservation laws in continuum mechanics. Nonlinear Analysis and Mechanics Vol. III, pp. 135–173, New York: Pitman, 1978.

    Google Scholar 

  25. Štědrý, M., Small time-periodic solutions to fully nonlinear telegraph equations in more spatial dimensions. Ann. Inst. Henri Poincaré 6 (3), 209–232 (1989).

    Google Scholar 

  26. Tartar, L., Compensated compactness and applications to partial differential equations. Research Notes in Math. 39, Nonlinear Analysis and Mechanics: Heriot-Watt-Symposium Vol. 4, pp. 136–211. New York: Pitman, 1975.

    Google Scholar 

  27. Vejvoda, O., et al., Partial differential equations: Time periodic solutions. Martinus Nijhoff Publ., 1982.

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  1. Institute of Mathematics, Czechoslovak Academy of Sciences, Prague

    Eduard Feireisl

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  1. Eduard Feireisl
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Communicated by H. Brezis

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Feireisl, E. Time-periodic solutions to quasilinear telegraph equations with large data. Arch. Rational Mech. Anal. 112, 45–62 (1990). https://doi.org/10.1007/BF00431722

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