Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

Uniform Attractor for A Non-autonomous Parabolic Equation with Nonlocal Diffusion and Delay

Download PDF (582.1 KB) PP. 50 - 61 Pub. Date: April 12, 2018

DOI: 10.22606/jaam.2018.32002

Author(s)

  • Miaomiao Wang
    School of Science, Hohai University, Nanjing, Jiangsu 210098, China
  • Weiwei Chang
    School of Science, Hohai University, Nanjing, Jiangsu 210098, China
  • Xiaojun Li*
    School of Science, Hohai University, Nanjing, Jiangsu 210098, China

Abstract

This paper is devoted to study the long-time behavior of non-autonomous parabolic equation with nonlocal diffusion and hereditary effect, where time symbol is translation bound in L2 loc(R;H−1( )) and L2 loc(R;L2( )), respectively. By the energy estimates and asymptotic priori estimates of solutions, we obtain the existence and regularity of uniform attractor for the family of processes corresponding to original systems, respectively.

Keywords

Uniform attractor; delay; nonlocal diffusion; non-autonomous parabolic equation; uniform !-limit compact.

References

[1] S.B. de Menezes, Remarks on weak solutions for a nonlocal parabolic problem, Int.J.Math.Math. Sciences 2006 (2006)1-10.

[2] S.M. Zheng, M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptotic Anal.45(2005)301-312.

[3] M. Chipot, B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity 3 (1999), 65-81.

[4] J. Garcia-Luengo, P.Marin-Rubio, J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv.Nonlinear Stud.13(2013)331-357.

[5] A. Andami Ovono, Asymptotic behavior for a diffusion equation governed by nonlocal interaction, Electronic J.Differential.Equations 17(2010)1555-1666.

[6] M. Anguiano, P.E. Kloeden, T. Lorenz, Asymptotic behaviour of nonlocal reaction diffusion equations, Nonlinear Anal.73(2010)3044-3057.

[7] T. Caraballo, M. Herrera-Cobos, P. Marin-Rubio, Long time behavior of a nonautonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal.121(2015)3-18.

[8] W.W. Chang, X.J. Li, Dynamical behavior of non-autonomous parablic equation with nonlocal diffusion, J.Sci.Tech.Univ.Henan(Nat.Sci.)(Chinese) 37(5)(2016)77-82.

[9] M. Chipot, L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Applicable Anal. 80(2001)279-315.

[10] J. Garcia-Luengo, P.Marin-Rubio, Reaction-diffusion equations with non-autonomous force in H?1 and delays under measurablity conditions on the driving delay term, J.Math.Anal.Appl., 417(2014) 80-95.

[11] H.M. Wei, X.Z. Li, M. Martcheva, An epidemic model of a vector-borne disease with direct transmis- sion and time delay, J.Math.Anal.Appl.2(342)(2008)895-908.

[12] V.V. Chepyzhov, M.I.Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Province, RI,2002.

[13] S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for nonautonomous 2D navier-stokes equations with nor- mal external forces,Discrete.Cont.Dyn.Syst.13(3)(2005)701-709.

[14] H.T. Song, C.K.Zhong, Attractors of non-autonmous reaction-diffusion equation in Lp, Nonlinear Anal.(68)2008,1890-1897.

[15] J.C. Robinson, Infinite-dimensional Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,Cambridge University Press, 2001.

[16] R. Temam, Infinite-dimension Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.