Honam Mathematical Journal (호남수학학술지)

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CONVERGENCE AND DECAY ESTIMATES FOR A NON-AUTONOMOUS DISPERSIVE-DISSIPATIVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS

  • Kim, Eun-Seok (Department of Mathematics, Chonnam National University, Institute for General Education, Sunchon National University, Department of Mathematics, Kunsan National University, Department of Advanced Transport Machinery Systems, Mokpo National University)
  • Received : 2022.03.24
  • Accepted : 2022.04.22
  • Published : 2022.06.25
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Abstract

This paper deals with the long - time behavior of global bounded solutions for a non-autonomous dispersive-dissipative equation with time-dependent nonlinear damping terms under the null Dirichlet boundary condition. By a new Lyapunov functional and Łojasiewicz-Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, which depends on the decay of the non-autonomous term g(x, t), when damping coefficients are integral positive and positive-negative, respectively.

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References

  1. R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal-TMA 53 (2003), no. 7-8, 1017-1039. https://doi.org/10.1016/S0362-546X(03)00037-3
  2. H. Di and Y. Shang, Global existence and asymptotic behavior of solutions for the double dispersive-dissipative wave equation with nonlinear damping and source terms, Bound. Value Probl., DOI: 10.1186/s13661-015-0288-6, 2015(29)(2015): 15 pages.
  3. J. Ferreira and M. Rojas-Medar, On global weak solutions of a nonlinear evolution equation in noncylindrical domain, Proceedings of the 9th International Colloquium on Differential Equations, D. Bainov (Ed.), VSP. (1999), 155-162.
  4. F. Gazzola and M. Squassina, Global solutions and finite time blowup for damped semilinear wave equations, Ann. I. H. Poincare-AN 23 (2006), no. 2, 185-207. https://doi.org/10.1016/j.anihpc.2005.02.007
  5. A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Diff. 9 (1999), no. 2, 95-124. https://doi.org/10.1007/s005260050133
  6. A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal. 26 (2001), no. 1, 21-36.
  7. A. Haraux, Systemes dynamiques dissipatifs et applications, Paris, Masson, 1991.
  8. I. B. Hassen and A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy, J. Funct. Anal. 260 (2011), no. 10, 2933-2963. https://doi.org/10.1016/j.jfa.2011.02.010
  9. I. B. Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, HAL (2009), Article ID: hal-00392110, 16 pages.
  10. I. B. Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dyn. Differ. Equ. 23 (2011), no. 2, 315-332. https://doi.org/10.1007/s10884-011-9212-7
  11. M. Hayes and G. Saccomandi, Finite amplitude transverse waves in special incompressible viscoelastic solids, J. Elasticity 59 (2000), no. 1-2, 213-225. https://doi.org/10.1023/A:1011081920910
  12. M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differ. Equations 144 (1998), no. 2, 302-312. https://doi.org/10.1006/jdeq.1997.3392
  13. Z. Jiao, Convergence of global solutions for some classes of nonlinear damped wave equations, (2013), arXiv preprint, arXiv: 1309.2364.
  14. W. Liu, Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms, Front. Math. China 5 (2010), no. 3, 555-574. https://doi.org/10.1007/s11464-010-0060-2
  15. A. E. H. Love, A treatise on the mathematical theory of elasticity, New York, Cambridge University Press, 2013.
  16. R. Quintanilla and G. Saccomandi, Spatial behavior for a fourth-order dispersive equation, Q. Appl. Math. 64 (2006), no. 3, 547-560. https://doi.org/10.1090/S0033-569X-06-01025-2
  17. P. Villaggio, Mathematical models for elastic structures, New York, Cambridge University Press, 2005.
  18. R. Xu, X. Zhao, and J. Shen, Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms, Appl. Math. Mech. 29 (2008), 259-262. https://doi.org/10.1007/s10483-008-0213-y
  19. R. Xu and Y. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation an high energy level, Int. J.Math. 23 (2012), no. 5, DOI: 10.1142/S0129167X12500607, 10 pages.
  20. H. Yassine and A. Abbas, Long-time stabilization of solutions to a nonautonomous semilinear viscoelastic equation, Appl. Math. Opt. 73 (2016), no. 2, 251-269. https://doi.org/10.1007/s00245-015-9301-9