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Attractors for a class of perturbed nonclassical diffusion equations with memory
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 |
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 [
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.
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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