DOI QR코드

DOI QR Code

UNIFORM ATTRACTORS FOR NON-AUTONOMOUS NONCLASSICAL DIFFUSION EQUATIONS ON ℝN

  • Received : 2012.04.04
  • Published : 2014.09.30

Abstract

We prove the existence of uniform attractors $\mathcal{A}_{\varepsilon}$ in the space $H^1(\mathbb{R}^N){\cap}L^p(\mathbb{R}^N)$ for the following non-autonomous nonclassical diffusion equations on $\mathbb{R}^N$, $$u_t-{\varepsilon}{\Delta}u_t-{\Delta}u+f(x,u)+{\lambda}u=g(x,t),\;{\varepsilon}{\in}(0,1]$$. The upper semicontinuity of the uniform attractors $\{\mathcal{A}_{\varepsilon}\}_{{\varepsilon}{\in}[0,1]}$ at ${\varepsilon}=0$ is also studied.

Acknowledgement

Supported by : Vietnam Ministry of Education and Training

References

  1. E. C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980), no. 3-4, 265-296.
  2. C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal. 73 (2010), no. 2, 399-412. https://doi.org/10.1016/j.na.2010.03.031
  3. C. T. Anh and T. Q. Bao, Dynamics of non-autonomous nonclassical diffusion equations on $\mathbb{R}^N$, Commun. Pure Appl. Anal. 11 (2012), no. 3, 1231-1252.
  4. G. Chen and C. K. Zhong, Uniform attractors for non-autonomous p-Laplacian equation, Nonlinear Anal. 68 (2008), no. 11, 3349-3363. https://doi.org/10.1016/j.na.2007.03.025
  5. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.
  6. J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
  7. J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys. 19 (1968), no. 4, 614-627. https://doi.org/10.1007/BF01594969
  8. H. Song, S. Ma, and C. K. Zhong, Attractors of non-autonomous reaction-diffusion equations, Nonlinearity 22 (2009), no. 3, 667-681. https://doi.org/10.1088/0951-7715/22/3/008
  9. H. Song and C. K. Zhong, Attractors of non-autonomous reaction-diffusion equations in Lp, Nonlinear Anal. 68 (2008), no. 7, 1890-1897. https://doi.org/10.1016/j.na.2007.01.059
  10. C. Sun, S. Wang, and C. K. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser. 23 (2007), no. 7, 1271-1280. https://doi.org/10.1007/s10114-005-0909-6
  11. C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymp. Anal. 59 (2008), no. 1-2, 51-81.
  12. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, Philadelphia, 1995.
  13. T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal. 14 (1963), 1-26. https://doi.org/10.1007/BF00250690
  14. C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Encyclomedia of Physics, Springer, Berlin, 1995.
  15. B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 179 (1999), no. 1, 41-52.
  16. S. Wang, D. Li, and C. K. Zhong, On the dynamic of a class of nonclassical parabolic equations, J. Math. Anal. Appl. 317 (2006), no. 2, 565-582. https://doi.org/10.1016/j.jmaa.2005.06.094
  17. H. Wu and Z. Zhang, Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term, Dyn. Syst. 26 (2011), no. 4, 391-400. https://doi.org/10.1080/14689367.2011.562185
  18. Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 2, 273-276. https://doi.org/10.1007/s102550200026

Cited by

  1. Strong global attractors for nonclassical diffusion equation with fading memory vol.2017, pp.1, 2017, https://doi.org/10.1186/s13662-017-1222-2
  2. Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity vol.31, 2016, https://doi.org/10.1016/j.nonrwa.2016.01.004
  3. Random Attractors of Stochastic Non-Autonomous Nonclassical Diffusion Equations with Linear Memory on a Bounded Domain vol.09, pp.11, 2018, https://doi.org/10.4236/am.2018.911085