Publication Cover
Dynamical Systems
An International Journal
Volume 37, 2022 - Issue 1
140
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Asymptotic analysis and upper semicontinuity to a system of coupled nonlinear wave equations

ORCID Icon, , &
Pages 29-55 | Received 27 Feb 2021, Accepted 26 Oct 2021, Published online: 26 Nov 2021

References

  • K. Agre and M.A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differ. Integral Equ. 19(11) (2006), pp. 1235–1270.
  • F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, Compt Rendus L'Académie Des Sci– Ser I – Math. 328(11) (1999), pp. 1015–1020.
  • F. Alabau, P. Cannarsa, and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ. 2(2) (2002), pp. 127–150.
  • F. Alabau-Boussouira, Z. Wang, and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM: COCV. 23(2) (2017), pp. 721–749.
  • R.G.C. Almeida and M.L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, Nonlinear Anal.: Real World Appl. 12(2) (2011), pp. 1023–1032.
  • C.O. Alves, M.M. Cavalcanti, V.N.D. Cavalcanti, M.A. Rammaha, and D. Toundykov, 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. – S 2(3) (2009), pp. 583–608.
  • A.V. Babin and M.I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, Elsevier Science, Amsterdam, 1992.
  • M.M. Cavalcanti, V.N. Domingos Cavalcanti, F.A. Falcão Nascimento, I. Lasiecka, and J.H. Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), pp. 1189–1206.
  • W. Charles, J.A. Soriano, F.A. Falcão Nascimento, and J.H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differ. Equ. 255(8) (2013), pp. 2267–2290.
  • M. Chipot, Elliptic equations: an introductory course, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Basel, 2009.
  • I.D. Chueshov, Introduction to the theory of infinite-dimensional dissipative systems, University lectures in contemporary mathematics, Acta Scientific Publishing House,Kharkiv, 2002.
  • I.D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, 2008.
  • I.D. Chueshov and I. Lasiecka, Von Karman evolution equations: Well-posedness and long time dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
  • S.M.S. Cordeiro, R.F.C. Lobato, and C.A. Raposo, Optimal polynomial decay for a coupled system of wave with past history, Open J. Math. Anal. 4(1) (2020), pp. 49–59.
  • M.J. Dos Santos, B. Feng, D.S. Almeida Júnior, and M.L. Santos, Global and exponential attractors for a nonlinear porous elastic system with delay term, Discrete Contin. Dyn. Syst. – B. 22(11) (2020), pp. 1–24.
  • M.J. Dos Santos, M.M. Freitas, A.J.A. Ramos, D.S. Almeida Júnior, and L.R.S. Rodrigues, Long-time dynamics of a nonlinear Timoshenko beam with discrete delay term and nonlinear damping, J. Math. Phys. 61 (2020), p. 061505.
  • M.J. Dos Santos, R.F.C. Lobato, S.M.S. Cordeiro, and A.C.B. Dos Santos, Quasi-stability and attractors for a nonlinear coupled wave system with memory, Bollettino Dell'Unione Matematica Italiana. 14 (2020), pp. 297–321.
  • M.M. Freitas, M.L. Santos, and J.A. Langa, Porous elastic system with nonlinear damping and sources terms, J. Differ. Equ. 264(4) (2018), pp. 2970–3051.
  • L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), pp. 385–394.
  • Y. Guo and M.A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal. 92(6) (2013), pp. 1101–1115.
  • Y. Guo and M.A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z Angewandte Math Phys. 64(3) (2013), pp. 621–658.
  • Y. Guo and M.A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc. 366 (2014), pp. 2265–2325.
  • J.K. Hale, Asymptotic behavior of dissipative systems, Mathematical surveys and monographs, American Mathematical Society, Providence, RI, 1988.
  • F. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert space, Ann. Differ. Equ. 1 (1985), pp. 43–56.
  • K. Jörgens, Das anfangswertproblem in großen für eine klasse nichtlinearer wellengleichungen, Math Z. 77(1) (1961), pp. 295–308.
  • S. Kesavan, Topics in functional analysis and applications, New Age International Pvt Ltd , New Delhi, 1989.
  • J.E. Lagnese, Boundary stabilization of thin plates, Society for Industrial and Applied Mathematics, Philadelphia,1989.
  • R.F.C. Lobato, S.M.S. Cordeiro, M.L. Santos, and D.S. Almeida Júnior, Optimal polynomial decay to coupled wave equations and its numerical properties, J. Appl. Math. 2014 (2014), p. 897080.
  • T.F. Ma and R.N. Monteiro, Singular limit and long-time dynamics of Bresse systems, SIAM J. Math. Anal. 49 (2017), pp. 2468–2495.
  • M.A. Rammaha and S. Sakuntasathien, Critically and degenerately damped systems of nonlinear wave equations with source terms, Appl. Anal. 89(8) (2010), pp. 1201–1227.
  • M.A. Rammaha and S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal.: Theor., Meth. Appl. 72(5) (2010), pp. 2658–2683.
  • A.J.A. Ramos, M.J. Dos Santos, M.M. Freitas, and D.S. Almeida Júnior, Existence of attractors for a nonlinear Timoshenko system with delay, J. Dyn. Differ. Equ. (2019), pp. 1–24.
  • M. Reed, Abstract non linear wave equations, Lecture Notes in Mathematicas, Springer-Verlag, Berlin, 1976.
  • D.L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl. 173(2) (1993), pp. 339–358.
  • B. Said-Houari, Global existence and decay of solutions of a nonlinear system of wave equations, Appl. Anal. 91(3) (2012), pp. 475–489.
  • M.L. Santos, M.P.C. Rocha, and S.C. Gomes, Polynomial stability of a coupled system of wave equations weakly dissipative, Appl. Anal. 86(10) (2007), pp. 1293–1302.
  • I. Segal, Non-linear semi-groups, Ann. Math. 78(2) (1963), pp. 339–364.
  • J. Simon, Compact sets in the space Lp(0,T;B), Annali Matematica Pura Appl 146(4) (1987), pp. 65–96.
  • R. Temam, Infinite-Dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Springer, New York, 2013.
  • S. Zheng, Nonlinear evolution equations, Chapman & Hall/CRC, Boca Raton, 2004.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.