On a system of nonlinear wave equations with Balakrishnan–Taylor damping

Article

Abstract

In this paper, we study the initial-boundary value problem for a coupled system of nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan–Taylor damping terms. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Also, we show that nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of stronger damping.

Mathematics Subject Classification (2000)

35L20 35L70 58G16 

Keywords

Balakrishnan–Taylor damping General decay Relaxation function Viscoelastic material Blow up 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alves C.O., Cavalcanti M.M., Domingos Cavalcanti V.N., Rammaha M. A., Toundyjov D.: On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete Contin. Dyn. Syst. Ser. S 2, 583–608 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Appleby J.A.D., Fabrizio M., Lazzri B., Reynolds D.W.: On exponential asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16, 1677–1694 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Balakrishnan, A.V., Taylor, L.W.: Distributed parameter nonlinear damping models for flight structures. In: Proceedings “Daming 89”, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB (1989)Google Scholar
  4. 4.
    Ball J.: Remarks on blow up and nonexistence theorems nonlinear evolution equations. Q. J. Math. Oxford 28, 473–486 (1977)CrossRefMATHGoogle Scholar
  5. 5.
    Barbu V., Lasiecka I., Rammaha M.A.: On nonlinear wave equations with degenerate damping and source terms. Trans. Am. Math. Soc. 357, 2571–2622 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bass, R.W., Zes, D.: Spillover nonlinearity, and flexible structures. In: Taylor, L.W. (ed.) The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065, pp. 1–14 (1991)Google Scholar
  7. 7.
    Berrimi S., Messaoudi S.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64, 2314–2331 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cannarsa P., Sforza D.: Integro-differential equations of hyperbolic type with positive definite kernels. J. Differ. Equ. 250, 4289–4335 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Prates Filho J.S., Soriano J.A.: Existence and uniform decay rates for viscoelastic problems with nonlocal boundary damping. Differ. Integral Equ. Appl. 14, 85–116 (2001)MATHMathSciNetGoogle Scholar
  10. 10.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Martinez P.: General decay rate estimates for viscoelastic dissipative system. Nonlinear Anal. Theory Methods Appl. 68, 177–193 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42, 1310–1324 (2003)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Lasiecka I.: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 236, 407–459 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Clark H.R.: Elastic membrane equation in bounded and unbounded domains. Electron. J. Qual. Theory Differ. Equ. 11, 1–21 (2002)Google Scholar
  14. 14.
    Glassey R.T.: Blow-up theorems for nonlinear wave equations. Math. Z. 132, 183–203 (1973)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Haraux A., Zuazua E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 100, 191–206 (1988)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Kalantarov V.K., ladyzhenskaya O.A.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type. J. Soviet Math. 10, 53–70 (1978)CrossRefGoogle Scholar
  17. 17.
    Kirchhoff G.: Vorlesungen über Mechanik. Tauber, Leipzig (1883)Google Scholar
  18. 18.
    Komornik V.: Exact Controllability and Stabilization The Multiplier Method Res. Appl. Math. vol. 36. Wiley-Masson, Pairs/Chichester (1994)Google Scholar
  19. 19.
    Kopackova M.: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Comment. Math. Univ. Carolin. 30, 713–719 (1989)MATHMathSciNetGoogle Scholar
  20. 20.
    Li M.R., Tsai L.Y.: Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. TMA 54, 1397–1415 (2003)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Medeiros, L.A., Milla Miranda, M.: Weak solutions for a system of nonlinear Klein-Gordon equations. Ann. Mat. Pura Appl. CXLVI, pp. 173–183 (1987)Google Scholar
  22. 22.
    Medeiros L.A., Menzala G.P.: On a mixed problem for a class of nonlinear Klein-Gordon equations. Acta Math. Hung. 52, 61–69 (1988)CrossRefMATHGoogle Scholar
  23. 23.
    Medjden M., Tatar T.-E.: On the wave equation with a temporal nonlocal term. Dyn. Syst. Appl. 16, 665–672 (2007)MATHMathSciNetGoogle Scholar
  24. 24.
    Mustafa M.I.: Well posedness and asymptotic behavior of coupled system of nonlinear viscoelastic equations. Nonlinear Anal. Real World Appl. 13, 452–463 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Pata V.: Exponential stability in linear viscoelasticity. Q. Appl. Math. 64, 499–513 (2006)MATHMathSciNetGoogle Scholar
  26. 26.
    Pitts D.R., Rammaha M.A.: Global existence and non-existence theorems for nonlinear wave equations. Indiana Univ. Math. J. 51(6), 1479–1509 (2002)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rammaha M.A.: The influence of damping and source terms on solutions of nonlinear wave equations. Bol. Soc. Parana. Mat. 25(3), 77–90 (2007)MATHMathSciNetGoogle Scholar
  28. 28.
    Segal L.E.: The global cauchy problem for relativistic scalar fields with power interactions. Bull. Soc. Math. France 91, 129–135 (1963)MATHMathSciNetGoogle Scholar
  29. 29.
    Tatar N., Zaraï A.: Exponential stability and blow up for a problem with Balakrishnan–Taylor damping. Demonstr. Math. XLIV 1, 67–90 (2011)Google Scholar
  30. 30.
    Vicente A.: Wave equation with acoustic/memory boundary condition. Boletim Soc. Parana. Mat. 27(3), 29–39 (2009)MATHMathSciNetGoogle Scholar
  31. 31.
    You Y.: Intertial manifolds and stabilization of nonlinear beam equations with Balakrishnan–Taylor damping. Abstr. Appl. Anal. 1, 83–102 (1996)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Zhang J.: On the standing wave in coupled non-linear Klein-Gordon equations. Math. Methods Appl. Sci. 26, 11–25 (2003)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Zarai A., Tatar N.-e.: Global existence and polynomial decay for a problem with Balakrishnan–Taylor damping. Arch. Math. (BRNO) 46, 157–176 (2010)MathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations