ASYMPTOTIC DYNAMIC OF THE NONCLASSICAL DIFFUSION EQUATION WITH TIME-DEPENDENT COEFFICIENT

Jing Wang; Qiaozhen Ma

doi:10.11948/20200055

2021 Volume 11 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(1): 445-463

Next Article Previous Article

Jing Wang, Qiaozhen Ma. ASYMPTOTIC DYNAMIC OF THE NONCLASSICAL DIFFUSION EQUATION WITH TIME-DEPENDENT COEFFICIENT[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 445-463. doi: 10.11948/20200055

Citation:

Jing Wang, Qiaozhen Ma. ASYMPTOTIC DYNAMIC OF THE NONCLASSICAL DIFFUSION EQUATION WITH TIME-DEPENDENT COEFFICIENT[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 445-463. doi: 10.11948/20200055

ASYMPTOTIC DYNAMIC OF THE NONCLASSICAL DIFFUSION EQUATION WITH TIME-DEPENDENT COEFFICIENT

Jing Wang¹,
Qiaozhen Ma^1, ,

1.
College of Mathematics and Statistics, Northwest Normal University, AnningDong Road, Lanzhou 730070, China

Corresponding author: Email address:maqzh@nwnu.edu.cn(Q. Ma)
Fund Project: The authors were supported by National Natural Science Foundation of China (11961059, 11561064, 11761062)

Abstract

Abstract

We study the asymptotic behavior of solutions for a nonclassical diffusion equation with polynomial growth condition of arbitrary order $ p\geq2 $ on bounded domain $ \Omega\subset\mathbb{R}^{N} $ with smooth boundary $ \partial\Omega $. Firstly, the existence and uniqueness of weak solution are obtained in the time-dependent space $ \mathcal{H}_{t} $ with the norm depending on time $ t $ explicitly. Then we establish the existence, regularity and asymptotic structure of the time-dependent global attractor.
- Nonclassical diffusion equation /
- time-dependent global attractor /
- polynomial growth of arbitrary order /
- asymptotic structure /
- regularity
MSC: 35B25, 35B40, 35B41, 37L30, 45K05

FullText(HTML)

References

[1]	E. A. Aifantis, On the problem of diffusion in solids, Acta Mechanica, 1980, 37, 265-296. doi: 10.1007/BF01202949 CrossRef Google Scholar
[2]	C. T. Anh, D. T. P. Thanh and N. D. Toan, Global attractors for nonclassical diffusion euqations with hereditary memory and a new class of nonlinearities, Ann. Polon. Math., 2017, 119, 1-21. doi: 10.4064/ap4015-2-2017 CrossRef Google Scholar
[3]	C. T. Anh, D. T. P. Thanh and N. D. Toan, Averaging of nonclasssical diffusion equations with memory and singularly oscillating forces, Z. Anal. Anwend., 2018, 37, 299-314. doi: 10.4171/ZAA/1615 CrossRef Google Scholar
[4]	C. T. Anh and N. D. Toan, Nonclassical diffusion equations on $\mathbb{R}.{N}$ with singularly oscillating external forces, Appl. Math. Lett., 2014, 38, 20-26. doi: 10.1016/j.aml.2014.06.008 CrossRef $\mathbb{R}.{N}$ with singularly oscillating external forces" target="_blank">Google Scholar
[5]	T. Caraballo, A. M. Marquez-Durán and F. Rivero, Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic, Discrete Contin. Dyn. Syst. Ser. B, 2017, 22, 1817-1833 Google Scholar
[6]	V. V. Chpyzhov and M. I. Vishik, Attractor for Equations of Mathematical Physics, Amer Mathematical Society, Providence, RI, 2002. Google Scholar
[7]	M. Conti, V. Danese, C. Giorgi and V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 2018, 140, 349-389. doi: 10.1353/ajm.2018.0008 CrossRef Google Scholar
[8]	M. Conti and V. Pata, On the regularity of global attractors, Discrete Contin. Dyn. Syst. Ser., 2009, 25, 1209-1217. doi: 10.3934/dcds.2009.25.1209 CrossRef Google Scholar
[9]	M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one, Appl. Math. Comput., 2015, 259, 32-44. Google Scholar
[10]	M. Conti, V. Pata and R. Temam, Attractors for the process on time-dependent spaces, applications to wave equation, J. Differential Equations, 2013, 255, 1254-1277. doi: 10.1016/j.jde.2013.05.013 CrossRef Google Scholar
[11]	T. Ding and Y. Liu, Time-dependent global attractor for the nonclassical diffusion equations, Appl. Anal., 2015, 94, 1439-1449. doi: 10.1080/00036811.2014.933475 CrossRef Google Scholar
[12]	K. Kuttler and E. C. Aifantis, Existence and uniqueness in nonclassical diffusion, Quart. Appl. Math., 1987, 45, 549-560. doi: 10.1090/qam/910461 CrossRef Google Scholar
[13]	J. L. Lions, Quelques méthodes de Résolutions Des Probléms Aus Limites Nonlinéaries, Dunod Gauthier-Villars, Paris, 1969. Google Scholar
[14]	J. L. Lions and E. Magenes, Non-homogeneous Boundary value Problem and Applications, Springer-Verlag, Berlin, 1972. Google Scholar
[15]	T. Liu and Q. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser. B, 2018, 23, 4595-4616. Google Scholar
[16]	Q. Ma, Y. Liu and F. Zhang, Global attractors in H_{1}(\mathbb{R}.{N})$ for nonclassical diffusion equation, Discrete Dyn. Nat. Soc., 2012, 2012. Article ID 672762. H_{1}(\mathbb{R}.{N})$ for nonclassical diffusion equation" target="_blank">Google Scholar
[17]	Q. Ma, J. Wang and T. Liu, Time-dependent asymptotic behavior of the solution for wave equations with linear memory, Comput. Math. Appl., 2018, 76, 1372-1387. doi: 10.1016/j.camwa.2018.06.031 CrossRef Google Scholar
[18]	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, 1-11. doi: 10.1186/s13661-015-0477-3 CrossRef Google Scholar
[19]	F. Meng and C. Liu, Necessary and sufficient condition for the existence of time-dependent global attractor and application, J. Math. Phys., 2017, 58, 1-9. Google Scholar
[20]	F. Meng, J. Wu and C. Zhao, Time-dependent global attractor for extensible berger equation, J. Math. Anal. Appl., 2019, 469, 1045-1069. doi: 10.1016/j.jmaa.2018.09.050 CrossRef Google Scholar
[21]	F. Meng, M. Yang and C. Zhong, Attractors for wave equations with nonlinear damping on time-dependent on time-dependent space, Discrete Contin. Dyn. Syst. Ser. B, 2015, 21, 205-225. doi: 10.3934/dcdsb.2016.21.205 CrossRef Google Scholar
[22]	V. Pata and M. Conti, Asymptotic structure of the attractor for process on time-dependent spaces, Nonlinear Anal. Real World Appl., 2014, 19, 1-10. doi: 10.1016/j.nonrwa.2014.02.002 CrossRef Google Scholar
[23]	J. G. Peter and M. E. Gurtin, On the theory of heat condition involving two temperatures, Z. Angew. Math. Phys., 1968, 19(4), 614-627. doi: 10.1007/BF01594969 CrossRef Google Scholar
[24]	F. D. Plinio, G. Duan and R. Temam, Time dependent attractor for the oscillon equation, Discrete Dyn. Nat. Soc., 2011, 29, 141-167. Google Scholar
[25]	J. C. Robinson, Infinite-Dimensional Dynamical Systens, Cambridge University press, 2011. Google Scholar
[26]	J. Simon, Compact sets in the space $L.{p}(0, t;b)$, Ann. Mat. Pura Appl., 1987, 146, 65-96. $L.{p}(0, t;b)$" target="_blank">Google Scholar
[27]	C. Sun, S. Wang and C. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser., 2007, 23(7), 1271-1280. doi: 10.1007/s10114-005-0909-6 CrossRef Google Scholar
[28]	C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptot. Anal., 2008, 59, 51-81. doi: 10.3233/ASY-2008-0886 CrossRef Google Scholar
[29]	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., 2019, 47, 309-325. doi: 10.1007/s10013-018-0320-0 CrossRef Google Scholar
[30]	S. Wang, D. Li and C. Zhong, On the dynamics of a class of nonclassical parabolic equations, J. Math. Anal. Appl., 2006, 317, 565-582. doi: 10.1016/j.jmaa.2005.06.094 CrossRef Google Scholar
[31]	X. Wang and Q. Ma, Asymptotic structure of time-dependent global attractors for the nonclassical diffusion equations, J. Sichuan Univ. Eng. Sci. Ed., 2016, 53, 508-511. Google Scholar
[32]	Y. Wang, Z. Zhu and P. Li, Regularity of pullback attractors for nonautonomous nonclassical diffusion equations, J. Math. Anal. Appl., 2018, 459, 16-31. doi: 10.1016/j.jmaa.2017.10.075 CrossRef Google Scholar
[33]	Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl. Ser., 2002, 18(2), 273-276. doi: 10.1007/s102550200026 CrossRef Google Scholar
[34]	Y. Xie, Q. Li and K. Zhu, Attractors for nonclassical diffusion equations with arbitrsry polynomial growth nonlinearity, Nonlinear Anal. Real World Appl., 2016, 31, 23-37. doi: 10.1016/j.nonrwa.2016.01.004 CrossRef Google Scholar
[35]	S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 2004, 3, 921-934. doi: 10.3934/cpaa.2004.3.921 CrossRef Google Scholar
[36]	F. Zhang and L. Bai, Attractors for the nonclassical diffusion equations of kirchhoff type with critical nonlinearity on unbounded domain, Dyn. Syst., 2019, 56, 1-24. Google Scholar
[37]	C. Zhong, M. 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, 2006, 223, 367-399. doi: 10.1016/j.jde.2005.06.008 CrossRef Google Scholar
[38]	K. Zhu and C. Sun, Pullback attractors for nonclassical diffusion equations with delays, J. Math. Phys., 2015, 56, 1-20. Google Scholar