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LONG TIME BEHAVIOR OF SOLUTIONS TO SEMILINEAR HYPERBOLIC EQUATIONS INVOLVING STRONGLY DEGENERATE ELLIPTIC DIFFERENTIAL OPERATORS

  • Luyen, Duong Trong (Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc Thang University and Faculty of Mathematics and Statistics Ton Duc Thang University) ;
  • Yen, Phung Thi Kim (Department of Mathematics Ha Noi University of Natural Resources and Environment)
  • Received : 2020.10.16
  • Accepted : 2021.03.10
  • Published : 2021.09.01

Abstract

The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate hyperbolic equation involving strongly degenerate elliptic differential operators. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

Keywords

Acknowledgement

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.13.

References

  1. C. T. Anh, Global attractor for a semilinear strongly degenerate parabolic equation on ℝN, NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 5, 663-678. https://doi.org/10.1007/s00030-013-0261-y
  2. A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. (9) 62 (1983), no. 4, 441-491 (1984).
  3. A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. R. Soc. Edinburgh, 116A (1990), 221-243. https://doi.org/10.1017/S0308210500031498
  4. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. English translation, North-Holland, 1992.
  5. V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. https://doi.org/10.1051/cocv:2002056
  6. I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183 pp. https://doi.org/10.1090/memo/0912
  7. D. Fall, Longtime dynamics of hyperbolic evolutionary equations in unbounded domains and lattice systems, ProQuest LLC, Ann Arbor, MI, 2005. https://scholarcommons.usf.edu/etd/2875
  8. E. Feireisl, Attractors for semilinear damped wave equations on R3, Nonlinear Anal. 23 (1994), no. 2, 187-195. https://doi.org/10.1016/0362-546X(94)90041-8
  9. N. I. Karachalios and N. M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on RN, J. Differential Equations 157 (1999), no. 1, 183-205. https://doi.org/10.1006/jdeq.1999.3618
  10. A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain, Appl. Math. Lett. 18 (2005), no. 7, 827-832. https://doi.org/10.1016/j.aml.2004.08.013
  11. A. E. Kogoj and E. Lanconelli, On semilinear ∆λ-Laplace equation, Nonlinear Anal. 75 (2012), no. 12, 4637-4649. https://doi.org/10.1016/j.na.2011.10.007
  12. A. E. Kogoj and S. Sonner, Attractors met X-elliptic operators, J. Math. Anal. Appl. 420 (2014), no. 1, 407-434. https://doi.org/10.1016/j.jmaa.2014.05.070
  13. D. T. Luyen, Two nontrivial solutions of boundary-value problems for semilinear ∆γ-differential equations, Math. Notes 101 (2017), no. 5-6, 815-823. https://doi.org/10.1134/S0001434617050078
  14. D. T. Luyen, Existence of nontrivial solution for fourth-order semilinear ∆γ-Laplace equation in ℝN, Electron. J. Qual. Theory Differ. Equ. 78 (2019), 1-12. https://doi.org/10.14232/ejqtde.2019.1.78
  15. D. T. Luyen, D. T. Huong, and L. T. H. Hanh, Existence of infinitely many solutions for ∆γ-Laplace problems, Math. Notes 103 (2018), no. 5-6, 724-736. https://doi.org/10.1134/S000143461805005X
  16. D. T. Luyen and N. M. Tri, Existence of solutions to boundary-value problems for similinear ∆γ differential equations, Math. Notes 97 (2015), no. 1-2, 73-84. https://doi.org/10.1134/S0001434615010101
  17. D. T. Luyen and N. M. Tri, Large-time behavior of solutions to degenerate damped hyperbolic equations, Siberian Math. J. 57 (2016), no. 4, 632-649. https://doi.org/10.1134/S0037446616040078
  18. D. T. Luyen and N. M. Tri, Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator, Ann. Polon. Math. 117 (2016), no. 2, 141-162. https://doi.org/10.4064/ap3831-3-2016
  19. D. T. Luyen and N. M. Tri, Existence of infinitely many solutions for semilinear degenerate Schrodinger equations, J. Math. Anal. Appl. 461 (2018), no. 2, 1271-1286. https://doi.org/10.1016/j.jmaa.2018.01.016
  20. D. T. Luyen and N. M. Tri, On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators, Complex Var. Elliptic Equ. 64 (2019), no. 6, 1050-1066. https://doi.org/10.1080/17476933.2018.1498086
  21. D. T. Luyen and N. M. Tri, Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator, Complex Var. Elliptic Equ. 65 (2020), no. 12, 2135-2150. https://doi.org/10.1080/17476933.2020.1730824
  22. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1
  23. G. Raugel, Global attractors in partial differential equations, in Handbook of dynamical systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. https://doi.org/10.1016/S1874-575X(02)80038-8
  24. J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. https://doi.org/10.1007/978-94-010-0732-0
  25. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-5037-9
  26. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-1-4684-0313-8
  27. P. T. Thuy and N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 279-298. https://doi.org/10.1007/s00030-011-0128-z
  28. P. T. Thuy and N. M. Tri, Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 1213-1224. https://doi.org/10.1007/s00030-012-0205-y
  29. N. M. Tri, Critical Sobolev exponent for degenerate elliptic operators, Acta Math. Vietnam. 23 (1998), no. 1, 83-94.
  30. N. M. Tri, Semilinear Degenerate Elliptic Differential Equations, Local and global theories, Lambert Academic Publishing, 2010, 271p.
  31. N. M. Tri, Recent Progress in the Theory of Semilinear Equations Involving Degenerate Elliptic Differential Operators, Publishing House for Science and Technology of the Vietnam Academy of Science and Technology, 2014, 380p.
  32. B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128 (1999), no. 1, 41-52. https://doi.org/10.1016/S0167-2789(98)00304-2
  33. H. Xiao, Asymptotic dynamics of plate equations with a critical exponent on unbounded domain, Nonlinear Anal. 70 (2009), no. 3, 1288-1301. https://doi.org/10.1016/j.na.2008.02.012