doi: 10.3934/dcdsb.2021313
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Strong pullback attractors for a nonclassical diffusion equation

1. 

College of Information Science and Technology, Donghua University, Shanghai 201620, China

2. 

Department of Mathematics

3. 

Institute for Nonlinear Science, Donghua University, Shanghai 201620, China

*Corresponding author: Yuming Qin

Received  June 2021 Revised  October 2021 Early access January 2022

Fund Project: The second author is supported by NNSF grant 12171082 and the Fundamental Research Funds for the Central Universities with contract number 2232021G-13

In this paper, we investigate the existence of pullback attractors for a nonclassical diffusion equation with Dirichlet boundary condition in $ H^2(\Omega)\cap H^1_0(\Omega) $. First, we prove the existence and uniqueness of strong solutions for a nonclassical diffusion equation. Then we prove the existence of pullback attractors in $ H^2(\Omega)\cap H^1_0(\Omega) $ by applying asymptotic a priori estimate method.

Citation: Xiaolei Dong, Yuming Qin. Strong pullback attractors for a nonclassical diffusion equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021313
References:
[1]

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

[2]

C. T. AnhD. T. P. Thanh and N. D. Toan, Global attractors for nonclassical diffusion equations with hereditary memory and a new class of nonlinearities, Ann. Polon. Math., 119 (2017), 1-21.  doi: 10.4064/ap4015-2-2017.  Google Scholar

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C. T. Anh and N. D. Toan, Pullback attractors for nonclassical diffusion equations in noncylindrical domains, Int. J. Math. Math. Sci., 2012 (2012), Art. ID 875913, 30 pp. doi: 10.1155/2012/875913.  Google Scholar

[4]

T. Caraballo and A. M. Marquez-Duran, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.  Google Scholar

[5]

T. Caraballo, A. M. Marquez-Duran and F. Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540021, 11 pp. doi: 10.1142/S0218127415400210.  Google Scholar

[6]

T. CaraballoA. M. Marquez-Duran and F. Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1817-1833.  doi: 10.3934/dcdsb.2017108.  Google Scholar

[7]

T. ChenZ. Chen and Y. B. Tang, Finite dimensionality of global attractors for a non-classical reaction diffusion equation with memory, Appl. Math. Lett., 25 (2012), 357-362.  doi: 10.1016/j.aml.2011.09.014.  Google Scholar

[8]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.  doi: 10.3934/cpaa.2005.4.115.  Google Scholar

[9]

V. V. Chepyzhov and A. Miranville, On trajectory and global attractors for semilinear heat equations with fading memory, Indiana Univ. Math. J., 55 (2006), 119-167.  doi: 10.1512/iumj.2006.55.2597.  Google Scholar

[10]

M. Conti and E. M. Marchini, A remark on nonclassical diffusion equations with memory, Appl. Math. Optim., 73 (2016), 1-21.  doi: 10.1007/s00245-015-9290-8.  Google Scholar

[11]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Methods Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[12]

M. ContiF. D. Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Comm. Pure Appl. Math., 19 (2020), 2035-2050.  doi: 10.3934/cpaa.2020090.  Google Scholar

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M. EfendievA. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect., 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar

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J. Garc$\acute{i}$a-Luengo and P. Mar$\acute{i}$n-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.  Google Scholar

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H. Harraga and M. Yebdri, Pullback attractors for a class of semilinear nonclassical diffusion equations with delay, Electron. J. Differential Equations, 2016 (2016), 1-33.   Google Scholar

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Z. Hu and Y. Wang, Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay, J. Math. Phys., 53 (2012), 072702, 17 pp. doi: 10.1063/1.4736847.  Google Scholar

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J. Lee and V. M. Toi, Attractors for nonclassical diffusion equations with dynamic boundary conditions, Nonlinear Anal., 195 (2020), 111737, 26 pp. doi: 10.1016/j.na.2019.111737.  Google Scholar

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Y. LiS. Wang and J. Wei, Finite fractal dimension of pullback attractors and application to non-autonomous reaction diffusion equations, Appl. Math. E-Notes, 10 (2010), 19-26.   Google Scholar

[20]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[21]

Y. Liu, Time-dependent global attractor for the nonclassical diffusion equations, Appl. Anal., 94 (2015), 1439-1449.  doi: 10.1080/00036811.2014.933475.  Google Scholar

[22]

Y. Liu and Q. Ma, Exponential attractor for a nonclassical diffusion equation, Electron. J. Differential Equations, 9 (2009), 1-7.   Google Scholar

[23]

G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.  doi: 10.1016/j.na.2010.03.023.  Google Scholar

[24]

Q. Ma, X. Wang and L. Xu, Existence and regularity of time-dependent global attractors for the nonclassical reaction diffusion equations with lower forcing term, Bound. Value Probl., 2016 (2016), Paper No. 10, 11 pp. doi: 10.1186/s13661-015-0513-3.  Google Scholar

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M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.  Google Scholar

[26]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ, Linear Inequalities, Birkhäuser, 2016.  Google Scholar

[27]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅱ, Linear inequalities, Birkhäuser, 2016. Google Scholar

[28]

Y. Qin, Analytic Inequalities and Their Applications in PDEs, Birkh$\ddot{a}$user, 2017. doi: 10.1007/978-3-319-00831-8.  Google Scholar

[29]

F. Rivero, Time dependent perturbation in a non-autonomous nonclassical parabolic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 209-221.  doi: 10.3934/dcdsb.2013.18.209.  Google Scholar

[30] J. C. Robinson, Infinite Dimentional Dynamical System, Cambridge University Press, 2001.   Google Scholar
[31]

C. SunS. Wang and C. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. Engl. Ser., 23 (2007), 1271-1280.  doi: 10.1007/s10114-005-0909-6.  Google Scholar

[32]

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

[33]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65.  doi: 10.1016/j.na.2005.04.034.  Google Scholar

[34]

D. T. P. Thanh and N. D. Toan, Existence and long-time behavior of solutions to a class of nonclassical diffusion equations with infinite delays, Vietnam J. Math., 47 (2019), 309-325.  doi: 10.1007/s10013-018-0320-0.  Google Scholar

[35]

N. D. Toan, Existence and long-time behavior of variational solutions to a class of nonclassical diffusion equations in noncylindrical domains, Acta Math. Vietnam., 41 (2016), 37-53.  doi: 10.1007/s40306-015-0120-5.  Google Scholar

[36]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[37]

G. Wang and C. Hu, Continuous dependence on a parameter of exponential attractors for nonclassical diffusion equations, Discrete Dyn. Nat. Soc., 1 (2020), Art. ID 1025457, 12 pp. doi: 10.1155/2020/1025457.  Google Scholar

[38]

S. WangD. 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

[39]

X. WangL. 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

[40]

X. Wang and C. Zhong, Attractors for the non-autonomous nonclassical diffusion equations with fáding memory, Nonlinear Anal., 71 (2009), 5733-5746.  doi: 10.1016/j.na.2009.05.001.  Google Scholar

[41]

Y. Wang, P. Li and Y. Qin, Upper semicontinuity of uniform attractors for nonclassical diffusion equations, Bound. Value Probl., 2017 (2017), Paper No. 84, 11 pp. doi: 10.1186/s13661-017-0816-7.  Google Scholar

[42]

Y. Wang and Y. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., 5 (2010), 022701, 12 pp. doi: 10.1063/1.3277152.  Google Scholar

[43]

Y. Wang and L. Wang, Trajectory attractors for nonclassical diffusion equations with fading memory, Acta Math. Sci. Ser., 33 (2013), 721-737.  doi: 10.1016/S0252-9602(13)60033-8.  Google Scholar

[44]

Y. Wang and C. Zhong, On the existence of pullback attractors for nonautonomous reaction diffusion equations, Dyn. Syst., 23 (2008), 1-16.  doi: 10.1080/14689360701611821.  Google Scholar

[45]

Y. WangZ. Zhu and P. Li, Regularity of pullback attractors for nonautonomous nonclassical diffusion equations, J. Math. Anal. Appl., 459 (2018), 16-31.  doi: 10.1016/j.jmaa.2017.10.075.  Google Scholar

[46]

H. Wu and Z. Zhang, Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term, Dyn. Syst., 26 (2011), 391-400.  doi: 10.1080/14689367.2011.562185.  Google Scholar

[47]

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

[48]

Y. Xie, J. Li and K. Zhu, Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth, Adv. Difference Equ., 2021 (2021), Paper No. 75, 17 pp. doi: 10.1186/s13662-020-03146-2.  Google Scholar

[49]

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

[50]

Y. Zhang and Q. Ma, Exponential attractors of the nonclassical diffusion equations with lower regular forcing term, International Journal of Modern Nonlinear Theory and Application, 3 (2014), 15-22.  doi: 10.4236/ijmnta.2014.31003.  Google Scholar

[51]

Y. Zhang, X. Wang and C. Gao, Strong global attractors for nonclassical diffusion equation with fading memory, Adv. Difference Equ., (2017), Paper No. 163, 14 pp. doi: 10.1186/s13662-017-1222-2.  Google Scholar

[52]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[53]

K. Zhu and C. Sun, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys., 56 (2015), 092703, 20 pp. doi: 10.1063/1.4931480.  Google Scholar

[54]

K. Zhu, Y. Xie and F. Zhou, Attractors for the nonclassical reaction diffusion equations on time-dependent spaces, Bound. Value Probl., 2020 (2020), Paper No. 95, 14 pp. doi: 10.1186/s13661-020-01392-7.  Google Scholar

[55]

K. ZhuY. XieF. Zhou and X. Li, Uniform attractors for the non-autonomous reaction diffusion equations with delays, Asymptot. Anal., 123 (2021), 263-288.  doi: 10.3233/ASY-201633.  Google Scholar

show all references

References:
[1]

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

[2]

C. T. AnhD. T. P. Thanh and N. D. Toan, Global attractors for nonclassical diffusion equations with hereditary memory and a new class of nonlinearities, Ann. Polon. Math., 119 (2017), 1-21.  doi: 10.4064/ap4015-2-2017.  Google Scholar

[3]

C. T. Anh and N. D. Toan, Pullback attractors for nonclassical diffusion equations in noncylindrical domains, Int. J. Math. Math. Sci., 2012 (2012), Art. ID 875913, 30 pp. doi: 10.1155/2012/875913.  Google Scholar

[4]

T. Caraballo and A. M. Marquez-Duran, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.  Google Scholar

[5]

T. Caraballo, A. M. Marquez-Duran and F. Rivero, Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1540021, 11 pp. doi: 10.1142/S0218127415400210.  Google Scholar

[6]

T. CaraballoA. M. Marquez-Duran and F. Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1817-1833.  doi: 10.3934/dcdsb.2017108.  Google Scholar

[7]

T. ChenZ. Chen and Y. B. Tang, Finite dimensionality of global attractors for a non-classical reaction diffusion equation with memory, Appl. Math. Lett., 25 (2012), 357-362.  doi: 10.1016/j.aml.2011.09.014.  Google Scholar

[8]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.  doi: 10.3934/cpaa.2005.4.115.  Google Scholar

[9]

V. V. Chepyzhov and A. Miranville, On trajectory and global attractors for semilinear heat equations with fading memory, Indiana Univ. Math. J., 55 (2006), 119-167.  doi: 10.1512/iumj.2006.55.2597.  Google Scholar

[10]

M. Conti and E. M. Marchini, A remark on nonclassical diffusion equations with memory, Appl. Math. Optim., 73 (2016), 1-21.  doi: 10.1007/s00245-015-9290-8.  Google Scholar

[11]

M. ContiE. M. Marchini and V. Pata, Nonclassical diffusion with memory, Math. Methods Appl. Sci., 38 (2015), 948-958.  doi: 10.1002/mma.3120.  Google Scholar

[12]

M. ContiF. D. Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Comm. Pure Appl. Math., 19 (2020), 2035-2050.  doi: 10.3934/cpaa.2020090.  Google Scholar

[13]

M. EfendievA. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect., 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar

[14]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[15]

J. Garc$\acute{i}$a-Luengo and P. Mar$\acute{i}$n-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.  Google Scholar

[16]

H. Harraga and M. Yebdri, Pullback attractors for a class of semilinear nonclassical diffusion equations with delay, Electron. J. Differential Equations, 2016 (2016), 1-33.   Google Scholar

[17]

Z. Hu and Y. Wang, Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay, J. Math. Phys., 53 (2012), 072702, 17 pp. doi: 10.1063/1.4736847.  Google Scholar

[18]

J. Lee and V. M. Toi, Attractors for nonclassical diffusion equations with dynamic boundary conditions, Nonlinear Anal., 195 (2020), 111737, 26 pp. doi: 10.1016/j.na.2019.111737.  Google Scholar

[19]

Y. LiS. Wang and J. Wei, Finite fractal dimension of pullback attractors and application to non-autonomous reaction diffusion equations, Appl. Math. E-Notes, 10 (2010), 19-26.   Google Scholar

[20]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[21]

Y. Liu, Time-dependent global attractor for the nonclassical diffusion equations, Appl. Anal., 94 (2015), 1439-1449.  doi: 10.1080/00036811.2014.933475.  Google Scholar

[22]

Y. Liu and Q. Ma, Exponential attractor for a nonclassical diffusion equation, Electron. J. Differential Equations, 9 (2009), 1-7.   Google Scholar

[23]

G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.  doi: 10.1016/j.na.2010.03.023.  Google Scholar

[24]

Q. Ma, X. Wang and L. Xu, Existence and regularity of time-dependent global attractors for the nonclassical reaction diffusion equations with lower forcing term, Bound. Value Probl., 2016 (2016), Paper No. 10, 11 pp. doi: 10.1186/s13661-015-0513-3.  Google Scholar

[25]

M. Marion, Attractors for reactions-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.  Google Scholar

[26]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅰ, Linear Inequalities, Birkhäuser, 2016.  Google Scholar

[27]

Y. Qin, Integral and Discrete Inequalities and Their Applications, Vol. Ⅱ, Linear inequalities, Birkhäuser, 2016. Google Scholar

[28]

Y. Qin, Analytic Inequalities and Their Applications in PDEs, Birkh$\ddot{a}$user, 2017. doi: 10.1007/978-3-319-00831-8.  Google Scholar

[29]

F. Rivero, Time dependent perturbation in a non-autonomous nonclassical parabolic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 209-221.  doi: 10.3934/dcdsb.2013.18.209.  Google Scholar

[30] J. C. Robinson, Infinite Dimentional Dynamical System, Cambridge University Press, 2001.   Google Scholar
[31]

C. SunS. Wang and C. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. Engl. Ser., 23 (2007), 1271-1280.  doi: 10.1007/s10114-005-0909-6.  Google Scholar

[32]

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

[33]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65.  doi: 10.1016/j.na.2005.04.034.  Google Scholar

[34]

D. T. P. Thanh and N. D. Toan, Existence and long-time behavior of solutions to a class of nonclassical diffusion equations with infinite delays, Vietnam J. Math., 47 (2019), 309-325.  doi: 10.1007/s10013-018-0320-0.  Google Scholar

[35]

N. D. Toan, Existence and long-time behavior of variational solutions to a class of nonclassical diffusion equations in noncylindrical domains, Acta Math. Vietnam., 41 (2016), 37-53.  doi: 10.1007/s40306-015-0120-5.  Google Scholar

[36]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[37]

G. Wang and C. Hu, Continuous dependence on a parameter of exponential attractors for nonclassical diffusion equations, Discrete Dyn. Nat. Soc., 1 (2020), Art. ID 1025457, 12 pp. doi: 10.1155/2020/1025457.  Google Scholar

[38]

S. WangD. 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

[39]

X. WangL. 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

[40]

X. Wang and C. Zhong, Attractors for the non-autonomous nonclassical diffusion equations with fáding memory, Nonlinear Anal., 71 (2009), 5733-5746.  doi: 10.1016/j.na.2009.05.001.  Google Scholar

[41]

Y. Wang, P. Li and Y. Qin, Upper semicontinuity of uniform attractors for nonclassical diffusion equations, Bound. Value Probl., 2017 (2017), Paper No. 84, 11 pp. doi: 10.1186/s13661-017-0816-7.  Google Scholar

[42]

Y. Wang and Y. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations, J. Math. Phys., 5 (2010), 022701, 12 pp. doi: 10.1063/1.3277152.  Google Scholar

[43]

Y. Wang and L. Wang, Trajectory attractors for nonclassical diffusion equations with fading memory, Acta Math. Sci. Ser., 33 (2013), 721-737.  doi: 10.1016/S0252-9602(13)60033-8.  Google Scholar

[44]

Y. Wang and C. Zhong, On the existence of pullback attractors for nonautonomous reaction diffusion equations, Dyn. Syst., 23 (2008), 1-16.  doi: 10.1080/14689360701611821.  Google Scholar

[45]

Y. WangZ. Zhu and P. Li, Regularity of pullback attractors for nonautonomous nonclassical diffusion equations, J. Math. Anal. Appl., 459 (2018), 16-31.  doi: 10.1016/j.jmaa.2017.10.075.  Google Scholar

[46]

H. Wu and Z. Zhang, Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term, Dyn. Syst., 26 (2011), 391-400.  doi: 10.1080/14689367.2011.562185.  Google Scholar

[47]

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

[48]

Y. Xie, J. Li and K. Zhu, Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth, Adv. Difference Equ., 2021 (2021), Paper No. 75, 17 pp. doi: 10.1186/s13662-020-03146-2.  Google Scholar

[49]

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

[50]

Y. Zhang and Q. Ma, Exponential attractors of the nonclassical diffusion equations with lower regular forcing term, International Journal of Modern Nonlinear Theory and Application, 3 (2014), 15-22.  doi: 10.4236/ijmnta.2014.31003.  Google Scholar

[51]

Y. Zhang, X. Wang and C. Gao, Strong global attractors for nonclassical diffusion equation with fading memory, Adv. Difference Equ., (2017), Paper No. 163, 14 pp. doi: 10.1186/s13662-017-1222-2.  Google Scholar

[52]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[53]

K. Zhu and C. Sun, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys., 56 (2015), 092703, 20 pp. doi: 10.1063/1.4931480.  Google Scholar

[54]

K. Zhu, Y. Xie and F. Zhou, Attractors for the nonclassical reaction diffusion equations on time-dependent spaces, Bound. Value Probl., 2020 (2020), Paper No. 95, 14 pp. doi: 10.1186/s13661-020-01392-7.  Google Scholar

[55]

K. ZhuY. XieF. Zhou and X. Li, Uniform attractors for the non-autonomous reaction diffusion equations with delays, Asymptot. Anal., 123 (2021), 263-288.  doi: 10.3233/ASY-201633.  Google Scholar

[1]

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