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September  2022, 27(9): 4995-5007. doi: 10.3934/dcdsb.2021261

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 Science, National University of Defense Technology, Changsha Hunan 410073, China

* Corresponding author: Yongqin Xie

Received  June 2021 Revised  September 2021 Published  September 2022 Early access  November 2021

Fund Project: The work was partly supported by NSFS grants 51578080, 11101053, 71471020, Postgraduate scientific research innovation project of Hunan Province (CX20210751)

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.

 

Correction: “College of Arts and Sciences” has been changed to "College of Science"; “Postgraduate scientific research innovation project of Hunan Province (CX20210751)” has been added to Fund Project. We apologize for any inconvenience this may cause.

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 and Continuous Dynamical Systems - B, 2022, 27 (9) : 4995-5007. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

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

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

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

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

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