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doi: 10.3934/dcdsb.2021261
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Attractors for a class of perturbed nonclassical diffusion equations with memory

Jianbo Yuan 1, , Shixuan Zhang 1, , Yongqin Xie 2,, and  Jiangwei Zhang 3,

1. 

School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha, Changsha Hunan 410114, China
2. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Changsha Hunan 410114, China
3. 

College of Arts and Sciences, National University of Defense Technology, Changsha Hunan 410073, China

* Corresponding author: Yongqin Xie

Received  June 2021 Revised  September 2021 Early access November 2021

Fund Project: The work was partly supported by NSFS grants 51578080, 11101053, 71471020

In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [7,14] as the perturbed parameters $ \nu = 0 $, but the nonlinearity $ f $ satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.

Keywords: The perturbed nonclassical diffusion equation, arbitrary polynomial growth, global attractor, upper semi-continuity.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35K57.
Citation: Jianbo Yuan, Shixuan Zhang, Yongqin Xie, Jiangwei Zhang. Attractors for a class of perturbed nonclassical diffusion equations with memory. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021261
References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[3]

C. T. Anh and N. D. Toan, Pullback attractors for nonclassical diffusion equations in noncylindrical domains, Internat. J. Math. & Math. Sci., 2012 (2012), 875913.  doi: 10.1155/2012/875913.  Google Scholar

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C. T. Anh and N. D. Toan, Existence and upper semicontinuity of uniform attractors in $H^1(R^N)$ for nonautonomous nonclassical difusion equations, Ann. Pol. Math., 111 (2014), 271-295.  doi: 10.4064/ap111-3-5.  Google Scholar

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P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

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D. Colton, Pseudo-parabolic equations in one space variable, J. Differ. Equations, 12 (1972), 559-565.  doi: 10.1016/0022-0396(72)90025-3.  Google Scholar

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M. Conti, F. Dell'Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Commun. Pur. Appl. Anal., 19 (2020), 2035-2050.  doi: 10.3934/cpaa.2020090.  Google Scholar

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S. L. Sobolev, On a new problems in mathematical physics, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50.   Google Scholar

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C. Sun, D. Dao and J. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyna. Syst., 6 (2008), 293-318.  doi: 10.1137/060663805.  Google Scholar

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C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asympt. Anal., 59 (2008), 51-81.  doi: 10.3233/ASY-2008-0886.  Google Scholar

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N. D. Toan, Uniform attractors of nonclassical diffusion equations lacking instantaneous damping on $\mathbb {R}^{N} $ with memory, Acta Appl. Math., 170 (2020), 789-822.  doi: 10.1007/s10440-020-00359-1.  Google Scholar

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F. Wang, P. Wang and Z. Yao, Approximate controllability of fractioanal paritial differential equation, Adv. Differ. Equ., 2015 (2015), 367, 10 pp. doi: 10.1186/s13662-015-0692-3.  Google Scholar

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L. Wang, Y. Wang and Y. Qin, Upper semicontinuity of attractors for nonclassical diffusion equations in $H^1(\mathbb{R}^3)$, Appl. Math.Comput., 240 (2014), 51-61.  doi: 10.1016/j.amc.2014.04.092.  Google Scholar

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S. Wang, D. Li and C. Zhong, On the dynamics of a class of nonclassical parabolic equations, J. Math. Anal. Appl., 317 (2006), 565-582.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

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X. Wang, L. Yang and C. Zhong, Attractors for the nonclassical diffusion equations with fading memory, J. Math. Anal. Appl., 362 (2010), 327-337.  doi: 10.1016/j.jmaa.2009.09.029.  Google Scholar

[19]

Y. Wang, P. Li and Y. Qin, Upper semicontinuity of uniform attractors for nonclassical diffusion equations, Bound. Value. Probl., 2017 (2017), 84.  doi: 10.1186/s13661-017-0816-7.  Google Scholar

[20]

Y. Wang and Y. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., 51 (2010), 022701.  doi: 10.1063/1.3277152.  Google Scholar

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Y. L. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. English Ser., 18 (2002), 273-276.  doi: 10.1007/s102550200026.  Google Scholar

[22]

Y. Xie, J. Li and K. Zhu, Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth, Adv. Differ. Equ., 2021 (2021), 75.  doi: 10.1186/s13662-020-03146-2.  Google Scholar

[23]

Y. Xie, Y. Li and Y. Zeng, Uniform attractors for nonclassical diffusion equations with memory, J. Func. Spac., 2016 (2016), Art. ID 5340489, 11 pp. doi: 10.1155/2016/5340489.  Google Scholar

[24]

Y. Xie, Q. Li and K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal., 31 (2016), 23-37.  doi: 10.1016/j.nonrwa.2016.01.004.  Google Scholar

[25]

Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space, J. Math. Phys., 53 (2012), 082703, 11 pp. doi: 10.1063/1.4746693.  Google Scholar

[26]

J. Zhang, Y. Xie, Q. Luo and Z. Tang, Asymptotic behavior for the semi-linear reaction diffusion equations with memory, Adv. Differ. Equ., 2019 (2019), 510.  doi: 10.1186/s13662-019-2399-3.  Google Scholar

[27]

K. Zhu, Y. Xie, F. Zhou and X. Li, Pullback attractors for the non-autonomous recation-diffusion equations in $\mathbb{R}^n$, J. Math. Phys., 60 (2019), 0032702.  doi: 10.1063/1.5040329.  Google Scholar

show all references

References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Anal., 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[3]

C. T. Anh and N. D. Toan, Pullback attractors for nonclassical diffusion equations in noncylindrical domains, Internat. J. Math. & Math. Sci., 2012 (2012), 875913.  doi: 10.1155/2012/875913.  Google Scholar

[4]

C. T. Anh and N. D. Toan, Existence and upper semicontinuity of uniform attractors in $H^1(R^N)$ for nonautonomous nonclassical difusion equations, Ann. Pol. Math., 111 (2014), 271-295.  doi: 10.4064/ap111-3-5.  Google Scholar

[5]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

[6]

D. Colton, Pseudo-parabolic equations in one space variable, J. Differ. Equations, 12 (1972), 559-565.  doi: 10.1016/0022-0396(72)90025-3.  Google Scholar

[7]

M. Conti, F. Dell'Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Commun. Pur. Appl. Anal., 19 (2020), 2035-2050.  doi: 10.3934/cpaa.2020090.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 1998.  Google Scholar

[9] J. C. Robinson, Infinite-Dimensional Dynamical Dystems, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[10]

R. E. Showalter, Sobolev equations for nonlinear dispersive systems, Appl. Anal., 7 (1978), 297-308.  doi: 10.1080/00036817808839200.  Google Scholar

[11]

S. L. Sobolev, On a new problems in mathematical physics, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50.   Google Scholar

[12]

C. Sun, D. Dao and J. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyna. Syst., 6 (2008), 293-318.  doi: 10.1137/060663805.  Google Scholar

[13]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asympt. Anal., 59 (2008), 51-81.  doi: 10.3233/ASY-2008-0886.  Google Scholar

[14]

N. D. Toan, Uniform attractors of nonclassical diffusion equations lacking instantaneous damping on $\mathbb {R}^{N} $ with memory, Acta Appl. Math., 170 (2020), 789-822.  doi: 10.1007/s10440-020-00359-1.  Google Scholar

[15]

F. Wang, P. Wang and Z. Yao, Approximate controllability of fractioanal paritial differential equation, Adv. Differ. Equ., 2015 (2015), 367, 10 pp. doi: 10.1186/s13662-015-0692-3.  Google Scholar

[16]

L. Wang, Y. Wang and Y. Qin, Upper semicontinuity of attractors for nonclassical diffusion equations in $H^1(\mathbb{R}^3)$, Appl. Math.Comput., 240 (2014), 51-61.  doi: 10.1016/j.amc.2014.04.092.  Google Scholar

[17]

S. Wang, D. Li and C. Zhong, On the dynamics of a class of nonclassical parabolic equations, J. Math. Anal. Appl., 317 (2006), 565-582.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

[18]

X. Wang, L. Yang and C. Zhong, Attractors for the nonclassical diffusion equations with fading memory, J. Math. Anal. Appl., 362 (2010), 327-337.  doi: 10.1016/j.jmaa.2009.09.029.  Google Scholar

[19]

Y. Wang, P. Li and Y. Qin, Upper semicontinuity of uniform attractors for nonclassical diffusion equations, Bound. Value. Probl., 2017 (2017), 84.  doi: 10.1186/s13661-017-0816-7.  Google Scholar

[20]

Y. Wang and Y. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., 51 (2010), 022701.  doi: 10.1063/1.3277152.  Google Scholar

[21]

Y. L. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. English Ser., 18 (2002), 273-276.  doi: 10.1007/s102550200026.  Google Scholar

[22]

Y. Xie, J. Li and K. Zhu, Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth, Adv. Differ. Equ., 2021 (2021), 75.  doi: 10.1186/s13662-020-03146-2.  Google Scholar

[23]

Y. Xie, Y. Li and Y. Zeng, Uniform attractors for nonclassical diffusion equations with memory, J. Func. Spac., 2016 (2016), Art. ID 5340489, 11 pp. doi: 10.1155/2016/5340489.  Google Scholar

[24]

Y. Xie, Q. Li and K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal., 31 (2016), 23-37.  doi: 10.1016/j.nonrwa.2016.01.004.  Google Scholar

[25]

Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space, J. Math. Phys., 53 (2012), 082703, 11 pp. doi: 10.1063/1.4746693.  Google Scholar

[26]

J. Zhang, Y. Xie, Q. Luo and Z. Tang, Asymptotic behavior for the semi-linear reaction diffusion equations with memory, Adv. Differ. Equ., 2019 (2019), 510.  doi: 10.1186/s13662-019-2399-3.  Google Scholar

[27]

K. Zhu, Y. Xie, F. Zhou and X. Li, Pullback attractors for the non-autonomous recation-diffusion equations in $\mathbb{R}^n$, J. Math. Phys., 60 (2019), 0032702.  doi: 10.1063/1.5040329.  Google Scholar

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