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Well-Posedness and Asymptotic Behavior for a Nonlinear Wave Equation

  • Fágner Dias ArarunaEmail author
  • Frederico de Oliveira Matias
  • Milton de Lacerda Oliveira
  • Shirley Maria Santos e Souza
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)

Abstract

We consider the initial boundary value problem for nonlinear damped wave equations of the form \(u^{\prime \prime }+M({\textstyle \int _{\Omega }}\left \vert (-\Delta )^{s}u\right \vert ^{2}dx)\Delta u+(-\Delta )^{\alpha }u^{\prime }=f,\) with Neumann boundary conditions. We prove global existence of solutions, when s ∈ [1∕2, 1] and α ∈ (0, 1], and we show that the energy of these ones decays exponentially, as t →. The uniqueness of solutions is also obtained when α ∈ [1∕2, 1].

Keywords

Wave equation Well-posedness Asymptotic behavior 

AMS Subject Classifications

35L70 35B40 74K10 

Notes

Acknowledgement

The author “F. D. Araruna” is partially supported by INCTMat, CAPES, CNPq (Brazil).

References

  1. 1.
    Aassila, M.: On quasilinear wave equation with strong damping. Funkcial. Ekvac. 41, 67–78 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bernstein, S.: Sur une classe d’equations functionelles aux derivées partielles. Isv. Acad. Nauk SSSR, Serv. Math. 4, 17–26 (1940)Google Scholar
  3. 3.
    Brito, E.H.: The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability. Appl. Anal. 13, 219–233 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carrier, G.F.: On the nonlinear vibration problem of the elastic string. Q. J. Appl. Math. 3, 157–165 (1945)CrossRefGoogle Scholar
  5. 5.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  6. 6.
    Cousin, A.T., Frota, C.L., Larkin, N.A., Medeiros, L.A.: On the abstract model of the Kirchhoff-Carrier equation. Commun. Appl. Anal. 1(3), 389–404 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cousin, A.T., Frota, C.L., Larkin, N.A.: Existence of global solutions and energy decay for the Carrier equation with dissipative term. Differential Integral Equation 12(4), 453–469 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Frota, C.L., Goldstein, J.A.: Some nonlinear wave equations with acoustic boundary conditions. J. Differ. Equ. 164, 92–109 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kirchhoff, G.: Volersunger über Mechanik. Tauber Leipzig, Leipzig (1883)Google Scholar
  10. 10.
    Larkin, N.A.: Global regular solution for the nonhomogeneous Carrier equation. Math. Probl. Eng. 8, 15–31 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: de la Penha, G.M., Medeiros, L.A. (eds.) Contemporary Development in Continuons Mechanics and Partial Differential Equations. North-Holland, London (1978)Google Scholar
  12. 12.
    Lions, J.-L., Magenes, E.: Problémes aux limites non homogénes et applications, vol. 1. Dunod, Paris (1968)zbMATHGoogle Scholar
  13. 13.
    Matos, M.P., Pereira, D.: On a hyperbolic equation with strong dissipation. Funkcial. Ekvac. 34, 303–331 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Medeiros, L.A., Milla Miranda, M.: Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order. Math. Appl. Comp. 6, 257–276 (1987)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Medeiros, L.A., Milla Miranda, M.: On a nonlinear wave equation with damping. Rev. Math. Univ. Complu. Madrid 3, 213–231 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Medeiros, L.A., Limaco, J., Menezes, S.B.: Vibrations of elastic string: mathematical aspects, part 1. J. Comput. Anal. Appl. 4(2), 91–127 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Medeiros, L.A., Limaco, J., Menezes, S.B.: Vibrations of elastic string: mathematical aspects, part 2. J. Comput. Anal. Appl. 4(3), 211–263 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mimouni, S., Benaissa, A., Amroun, N.-E.: Global existence and optimal decay rate of solutions for the degenerate quasilinear wave equation with a strong dissipation. Appl. Anal. 89(6), 815–831 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nishihara, K.: Degenerate quasilinear hyperbolic equation with strong damping. Funkcial. Ekvac. 27(1), 125–145 (1984)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nishihara, K., Yamada, Y.: On global solutions of some degenerate quasi-linear hyperbolic equations with dissipative terms. Funkcial. Ekvac. 33(1), 151–159 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ono, K.: On decay properties of solutions for degenerate strongly damped wave equations of Kirchhoff type. J. Math. Anal. Appl. 381(1), 229–239 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ono, K.: On sharp decay estimates of solutions for mildly degenerate dissipative wave equations of Kirchhoff type. Math. Methods Appl. Sci. 34(11), 1339–1352 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Park, J.Y., Bae, J.J., Jung, I.H.: On existence of global solutions for the carrier model with nonlinear damping and source terms. Appl. Anal. 77(3–4), 305–318 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pohozhaev, S.I.: On a class of quasilinear hyperbolic equations. Math. USSR Sbornik 25, 145–158 (1975)CrossRefGoogle Scholar
  25. 25.
    Vasconcelos, C.F., Teixeira, L.M.: Strong solution and exponential decay for a nonlinear hyperbolic equation. Appl. Anal. 55, 155–173 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yamada, Y.: On some quasilinear wave equations with dissipative terms. Nagoya Math. J. 87, 17–39 (1982)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zuazua, E.: Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1, 161–185 (1988)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Fágner Dias Araruna
    • 1
    Email author
  • Frederico de Oliveira Matias
    • 1
  • Milton de Lacerda Oliveira
    • 1
  • Shirley Maria Santos e Souza
    • 1
  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

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