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Recent results on semi-linear hyperbolic problems in bounded domains

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Partial Differential Equations

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Bibliography

  1. R.R. ADAMS, Sobolev spaces, Academic press, New York (1975).

    MATH  Google Scholar 

  2. L. AMERIO, G. PROUSE, Uniqueness and almost periodicity theorems for a non-linear wave equation, Atti Accad.Naz. Lincei Rend. Cl.Sci.Fis.Mat.Natur. 46 (1969), 1–8.

    MathSciNet  MATH  Google Scholar 

  3. L. AMERIO, G. PROUSE, Abstract almost periodic functions and functional equations, Van Nostrand, New-York (1971).

    Book  MATH  Google Scholar 

  4. A.V. BABIN, M.I. VISHIK, Regular attractors of semi-groups and evolution equations, J. Math. Pures et Appl. 62 (1983), 44–491.

    MathSciNet  MATH  Google Scholar 

  5. M. BIROLI, Bounded or almost-periodic solutions of the non linear vibrating membrane equation, Ricerche Mat. 22 (1973). 190–202.

    MathSciNet  MATH  Google Scholar 

  6. M. BIROLI, A. HARAUX, Asymptotic behavior for an almost periodic, strongly dissipative wave equation, J.Diff.Eq. 38, 3 (1980), 422–440.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. BREZIS, Problèmes unilatéraux, J.Math. Pure et Appl. 51 (1972), 1–168.

    MathSciNet  MATH  Google Scholar 

  8. H. BREZIS, J.M. CORON, L. NIRENBERG, Free vibrations for a non linear wave equation and a theorem of P. Rabinowitz, C.P.A. M. 33 (1980), 667–689.

    MathSciNet  MATH  Google Scholar 

  9. T. CAZENAVE, A. HARAUX, Propriétés oscillatoires des solutions de certaines equations des ondes semi-linéaires, C.R.A.S. Paris, 298 (1984), 449–452.

    MathSciNet  MATH  Google Scholar 

  10. T. CAZENAVE, A. HARAUX, Oscillatory phenomena associated to semi-linear wave equations in one spatial dimension. Trans. A.M.S., in press.

    Google Scholar 

  11. T. CAZENAVE, A. HARAUX, On the nature of free oscillations associated with some semilinear wave equations, in "Non-linear partial differential equations and their applications, College de France Seminar", vol. 7(H-Brezis & J.L. Lions Editors), Research Notes in Math. no 122, Pitman (1984), 59–79.

    Google Scholar 

  12. T. CAZENAVE, A. HARAUX, Some oscillatory properties of the wave equation in several space dimensions, to appear.

    Google Scholar 

  13. T. CAZENAVE, A. HARAUX, L. VAZQUEZ, F.B. WEISSLER, Nonlinear effects in the wave equation with a cubic restoring force, to appear.

    Google Scholar 

  14. F.A. FICKEN, B.A. FLEISMANN, Initial value problems and time-periodic solutions for a nonlinear wave equation, C.P.A.M. 10,3 (1957), 331–356.

    MathSciNet  MATH  Google Scholar 

  15. J.M. GHIDAGLIA, R. TEMAM, Attractors for damped nonlinear hyperbolic equations, to appear in J.Math. Pure et Appl..

    Google Scholar 

  16. J.K. HALE, Asymptotic behavior and dynamics in infinite dimensions, Nonlinear Differential Equations (Hale & Martines-Amores editors), Research Notes in Math. no 132, Pitman (1985), 1–42.

    Google Scholar 

  17. A. HARAUX, Nonlinear evolution equations: Global behavior of solutions, Lecture Notes in Math. no 841. Springer (1981).

    Google Scholar 

  18. A. HARAUX, Almost periodic forcing for a wave equation with a nonlinear, local damping term. Proc. Roy. Soc. Edinburgh, 94 A(1983), 195–212.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. HARAUX, Dissipativity in the sense of Levison for a class of second order nonlinear evolution equations. Nonlinear Analysis, T.M.A. 6,11 (1982), 1207–1220.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. HARAUX, Two remarks on dissipative hyperbolic problems, in "Nonlinear partial differential equations and their applications, College de France Seminar", vol. 7 (H. Brezis & J.L. Lions editors), Research Notes in Math. no 122, Pitman (1984), 161–179.

    Google Scholar 

  21. A. HARAUX, Non-resonance for a strongly dissipative wave equation in higher dimensions, Manuscripta Math. 53 (1985), 145–166.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. HARAUX, Propriétés d'oscillation des solutions de l'équation des ondes avec conditions de Dirichlet au bord, Séminaire Bony-Sjostrand-Meyer 1985, Exp. no 9.

    Google Scholar 

  23. A. HARAUX, A new characterization of weak solutions to the damped wave equation, Publ. Lab. d'analyse numérique no 85039 (1985), 16p.

    Google Scholar 

  24. A. HARAUX, Semi-linear hyperbolic problems in bounded domains, to appear in "Mathematical reports", J. Dieudonné Editor.

    Google Scholar 

  25. A. HARAUX, Nonlinear vibrations and the wave equation, Textos e Notas (L.A. Medeiros, Editor), to appear.

    Google Scholar 

  26. A. HARAUX, V. KOMORNIK, Anharmonic Fourier and the wave equation. Rev.Mat. Ibero-Americana, in press.

    Google Scholar 

  27. J.L. LIONS, W.A. STRAUSS, Some non-linear evolution equations Bull.Soc.Math. France 93 (1965), 43–96.

    MathSciNet  MATH  Google Scholar 

  28. P.A. MARCATI, Decay and Stability for nonlinear hyperbolic equations, J.Diff.Eq. 55, 1 (1984), 30–58.

    Article  MathSciNet  MATH  Google Scholar 

  29. P.A. MARCATI, Stability for second order abstract evolution equations, Nonlinear Analysis, T.M.A. 8,3 (1984), 237–252.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. NAKAO, On boundedness, periodicity and almos periodicity of solutions of some nonlinear partial differential equations, J. Diff.Eq. 19, (1975), 371–385.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. NAKAO, Asymptotic stability of the bounded or almost periodic solution of the wave equation with a nonlinear dissipative term, J.Math.Anal.Appl. 58 (1977), 336–343.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. NAKAO, A difference inequality and its applications to nonlinear evolution equations, J.Math.Soc.Japan 30, 4 (1978), 747–762.

    Article  MathSciNet  MATH  Google Scholar 

  33. G. PRODI, Soluzioni periodiche di equazioni a derivati parziali di tipo iperbolico non lineari, Ann.Mat. Pura Appl. 42 (1956) 25–49.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. PRODI, Soluzioni periodiche della equazione delle onde contermine dissipativo non lineare, Rend.Sem.Mat.Univ. Padova 36 (1966), 3749.

    MathSciNet  Google Scholar 

  35. G. PROUSE, Soluzioni quasi-periodiche della equazione delle onde con termine dissipativo non lineare, I, II, III, IV, Rend. Accad.Naz.Lincei 38,39 (1965).

    Google Scholar 

  36. P.H. RABINOWITZ, periodic solutions of nonlinear hyperbolic partial differential equations, C.P.A.M. 20 (1967), 145–205.

    MathSciNet  MATH  Google Scholar 

  37. P.H. RABINOWITZ, Free vibrations for a semi-linear wave equation, C.P.A.M. 31 (1978), 31–68.

    MathSciNet  MATH  Google Scholar 

  38. Y. YAMADA, On the decay of solutions for some nonlinear evolution equations of second order, Nagoya Math.J. 73 (1979), 69–98.

    Article  MathSciNet  MATH  Google Scholar 

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Fernando Cardoso Djairo G. de Figueiredo Rafael Iório Orlando Lopes

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© 1988 Springer-Verlag

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Haraux, A. (1988). Recent results on semi-linear hyperbolic problems in bounded domains. In: Cardoso, F., de Figueiredo, D.G., Iório, R., Lopes, O. (eds) Partial Differential Equations. Lecture Notes in Mathematics, vol 1324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100787

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  • DOI: https://doi.org/10.1007/BFb0100787

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