Skip to main content
SpringerLink
Log in

Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

For many evolution problems, a basic question is to establish convergence to equilibrium for globally defined solutions. This type of result is well known for the semilinear wave equation with linear dissipation. In this paper, we are concerned with the asymptotic behavior of global and bounded solutions of the following semilinear wave equation

$$u_{tt}+|u_t|^{\alpha}u_t-\Delta u=f(x,u) \quad {\rm in }\, \mathbb{R}_+ \times \Omega$$

with homogeneous Dirichlet boundary conditions and initial conditions. Here, α ≥ 0, \({\Omega \subset \mathbb{R}^{N}\, (N\geq1)}\) is a bounded domain with sufficiently smooth boundary and \({f:\Omega\times\mathbb{R}\longrightarrow \mathbb{R}}\) is analytic in the second variable, uniformly with respect to the first one. In this paper, we suppose that the set of stationary solutions is compact and we prove convergence of global and bounded solutions to an equilibrium, for some small value of α depending on the nonlinearity f. The case α = 0 corresponds to the wave equation with linear dissipation which is solved by Haraux and Jendoubi (Calc Var Partial Differ Equ 9:95–124, 1999).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Biroli M.: Bounded or almost-periodic solutions of the nonlinear membrane equation. Ricerche Mat 22, 190–202 (1973)

    MATH  MathSciNet  Google Scholar 

  2. Biroli M., Haraux A.: Asymptotic behavior for an almost periodic, strongly dissipative wave equation. J. Diff. Eq. 38(3), 422–440 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chergui L.: Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity. J. Dynam. Diff. Equa. 20, 643–652 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Haraux A.: Comportement à l’infini pour une équation d’ondes non-linéaire dissipative. (French). C. R. Acad. Sci. Paris S. A-B 287(7), A507–A509 (1978)

    MathSciNet  Google Scholar 

  5. Haraux A.: Comportement à l’infini pour certains systèmes dissipatifs non-linéaires. (French) Proc. Roy. Soc. Edinburgh Sect. 84A(3-4), 213–234 (1979)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  7. Haraux A.: A new characterization of weak solutions to the damped wave equation. Funkc. Eqvac. 31, 471–482 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Haraux A., Jendoubi M.A.: Convergence of bounded weak solutions of the wave equation with dissipation and analytic non linearity. Calc. Var. Partial Differential Equations 9, 95–124 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Haraux A., Jendoubi M.A.: On the convergence of global and bounded solution of some evolution equations. J. Evol. Equ. 7(3), 449–470 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Haraux A., Zuazua E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Rational Mech. Anal. 100(2), 191–206 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jendoubi M.A.: Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differential Equations 144, 302–312 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jendoubi M.A., Poláčik P.: Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. Roy. Soc. Edin. 133A, 1137–1153 (2003)

    Article  Google Scholar 

  13. S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S \({\sharp}\) 117, Les équations aux dérivées partielles, 1963.

  14. S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S. Bures-sur-Yvette, 1965.

  15. Nakao M.: Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term. J. Math. Anal. Appl. 58(2), 336–343 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  16. Simon L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. of Math. 118, 525–571 (1983)

    Article  MathSciNet  Google Scholar 

  17. L. Simon, Theorems on regularity and singularity of energy minimizing maps Lect. in Math., ETH Zürich, Birkhäuser (1996).

  18. Zuazua E.: Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Anal. 1(2), 161–185 (1988)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Bizerte, 7021, Jarzouna, Bizerte, Tunisia

    L. Chergui

  2. Institut Préparatoire aux Etudes d’Ingénieurs d’Elmanar, Campus universitaire, BP 244, 2092, Elmanar 2, Tunis, Tunisia

    L. Chergui

Authors
  1. L. Chergui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Chergui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chergui, L. Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity. J. Evol. Equ. 9, 405–418 (2009). https://doi.org/10.1007/s00028-009-0016-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-009-0016-8

Keywords

Search

Navigation