대한수학회지 (Journal of the Korean Mathematical Society)

  • 제61권4호
  • /
  • Pages.797-836
  • /
  • 2024
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ASYMPTOTIC BEHAVIOR FOR STRONGLY DAMPED WAVE EQUATIONS ON ℝ3 WITH MEMORY

  • Xuan-Quang Bui (Faculty of Fundamental Sciences Phenikaa University) ;
  • Duong Toan Nguyen (Faculty of Mathematics and Natural Sciences Haiphong University) ;
  • Trong Luong Vu (VNU-University of Education Vietnam National University, Department of Mathematics FPT University)
  • 투고 : 2023.10.09
  • 심사 : 2024.02.05
  • 발행 : 2024.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We consider the following strongly damped wave equation on ℝ3 with memory utt - αΔut - βΔu + λu - ∫0∞ κ'(s)∆u(t - s)ds + f(x, u) + g(x, ut) = h, where a quite general memory kernel and the nonlinearity f exhibit a critical growth. Existence, uniqueness and continuous dependence results are provided as well as the existence of regular global and exponential attractors of finite fractal dimension.

키워드

과제정보

This work was completed while the authors were visiting the Vietnam Institute of Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for its hospitality.

참고문헌

  1. V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on ℝ3, Discrete Contin. Dynam. Systems 7 (2001), no. 4, 719-735. https://doi.org/10.3934/dcds.2001.7.719
  2. S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal. 20 (1999), no. 3-4, 263-277.
  3. V. V. Chepyzhov, M. Conti, and V. Pata, Averaging of equations of viscoelasticity with singularly oscillating external forces, J. Math. Pures Appl. (9) 108 (2017), no. 6, 841-868. https://doi.org/10.1016/j.matpur.2017.05.007
  4. V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), no. 3-4, 251-273.
  5. M. C. Conti, V. Pata, and M. Squassina, Strongly damped wave equations on ℝ3 with critical nonlinearities, Commun. Appl. Anal. 9 (2005), no. 2, 161-176.
  6. C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297-308. https://doi.org/10.1007/BF00251609
  7. F. Dell'Oro and V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity 24 (2011), no. 12, 3413-3435. https://doi.org/10.1088/0951-7715/24/12/006
  8. F. Dell'Oro and V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Anal. 75 (2012), no. 14, 5723-5735. https://doi.org/10.1016/j.na.2012.05.019
  9. P. Ding and Z. Yang, Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on ℝN, Commun. Pure Appl. Anal. 20 (2021), no. 3, 1059-1076. https://doi.org/10.3934/cpaa.2021006
  10. F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. II, Russ. J. Math. Phys. 16 (2009), no. 1, 61-73. https://doi.org/10.1134/S1061920809010038
  11. F. Di Plinio, V. Pata, and S. V. Zelik, On the strongly damped wave equation with memory, Indiana Univ. Math. J. 57 (2008), no. 2, 757-780. https://doi.org/10.1512/iumj.2008.57.3266
  12. A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential attractors for dissipative evolution equations, RAM: Research in Applied Mathematics, 37, Masson, Paris, 1994.
  13. S. Gatti, A. Miranville, V. Pata, and S. Zelik, Attractors for semi-linear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math. 38 (2008), no. 4, 1117-1138. https://doi.org/10.1216/RMJ-2008-38-4-1117
  14. M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution equations, semigroups and functional analysis (Milano, 2000), 155-178, Progr. Nonlinear Differential Equations Appl., 50, Birkhauser, Basel, 2002.
  15. J.-L. Lions, Quelques methodes de resolution des probl'emes aux limites non lineaires, Dunod, Paris, 1969.
  16. V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001), no. 2, 505-529.
  17. R. Temam, Navier-Stokes equations and nonlinear functional analysis, second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995. https://doi.org/10.1137/1.9781611970050
  18. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, second edition, Applied Mathematical Sciences, 68, Springer, New York, 1997. https://doi.org/10.1007/978-1-4612-0645-3
  19. N. D. Toan, Time-dependent global attractors for strongly damped wave equations with time-dependent memory kernels, Dyn. Syst. 37 (2022), no. 3, 466-492. https://doi.org/10.1080/14689367.2022.2072710
  20. Z. Yang and Z. Liu, Global attractor for a strongly damped wave equation with fully supercritical nonlinearities, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 2181-2205. https://doi.org/10.3934/dcds.2017094
  21. S. Zhou, Global attractor for strongly damped nonlinear wave equations, Funct. Differ. Equ. 6 (1999), no. 3-4, 451-470.
  22. S. Zhou, Attractors for strongly damped wave equations with critical exponent, Appl. Math. Lett. 16 (2003), no. 8, 1307-1314. https://doi.org/10.1016/S0893-9659(03)90134-0