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Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  18 November 2021

Yuming Qin  and
Bin Yang
Yuming Qin
Affiliation:
Department of Mathematics, Institute for Nonlinear Science, Donghua University, Shanghai 201620, People's Republic of China [email protected]
Bin Yang
Affiliation:
College of Information Science and Technology, Donghua University, Shanghai 201620, People's Republic of China [email protected]
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Abstract

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

Keywords

Non-autonomous nonclassical diffusion equationstime-dependent pullback attractorsregularity

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35B41: Attractors 35B65: Smoothness and regularity of solutions 35K57: Reaction-diffusion equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 6 , December 2022 , pp. 1533 - 1550
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Adams, R. A. and Fournier, J. F.. Sobolev spaces (America: Academic Press, 2003).Google Scholar
Anguiano, M., Kloeden, P. E. and Lorenz, T.. Asymptotic behaviour of nonlocal reaction-diffusion equations. Nonlinear Anal. TMA 73 (2010), 3044–3057.CrossRefGoogle Scholar
Caballero, R., Marín-Rubio, P. and Valero, J.. Existence and characterization of attractors for a nonlocal reaction-diffusion equation with an energy functional. J. Dyn. Differ. Equ. (2021), 1–38. DOI: 10.1007/s10884-020-09933-5.Google Scholar
Caraballo, T., Herrera-Cobos, M. and Marín-Rubio, P.. Long-time behaviour of a nonautonomous parabolic equation with nonlocal diffusion and sublinear terms. Nonlinear Anal. 121 (2015), 3–18.CrossRefGoogle Scholar
Caraballo, T., Herrera-Cobos, M. and Marín-Rubio, P.. Robustness of non-autonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness. Discrete Contin. Dyn. Syst. 23 (2016), 1011–1036.Google Scholar
Caraballo, T., Herrera-Cobos, M. and Marín-Rubio, P.. Time-dependent attractors for nonautonomous nonlocal reaction-diffusion equations. Proc. R. Soc. Edinburgh A 148 (2018), 957–981.CrossRefGoogle Scholar
Chepyzhov, V. V. and Vishik, M. I.. Attractor for equations of mathematical physics (America: American Mathematical Society, 2002).Google Scholar
Chipot, M. and Lovat, B.. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. TMA 30 (1997), 4619–4627.CrossRefGoogle Scholar
Chipot, M. and Lovat, B.. On the asymptotic behaviour of some nonlocal problems. Positivity 3 (1999), 65–81.CrossRefGoogle Scholar
Chipot, M. and Molinet, L.. Asymptotic behaviour of some nonlocal diffusion problems. Appl. Anal. 80 (2001), 279–315.Google Scholar
Chipot, M., Valente, V. and Vergara-Caffarelli, G.. Remarks on a nonlocal problem involving the Dirichlet energy. Rend. Semin. Mat. Univ. Padova 110 (2003), 199–220.Google Scholar
Chipot, M. and Zheng, S.. Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 45 (2005), 301–312.Google Scholar
Conti, M. and Pata, V.. On the regularity of global attractors. Discrete Contin. Dyn. Syst. 25 (2009), 1209–1217.CrossRefGoogle Scholar
Conti, M., Pata, V. and Temam, R.. Attractors for the process on time-dependent spaces, applications to wave equation. J. Differ. Equ. 255 (2013), 1254–1277.CrossRefGoogle Scholar
Evans, L. C.. Partial differential equations (America: American Mathematical Society, 1998).Google Scholar
Lions, J. L.. Quelques méthodes de résolutions des problems aus limites nonlinéaries (France: Dunod Gauthier-Villars, 1969).Google Scholar
Lions, J. L. and Magenes, E.. Non-homogeneous boundary value problem and applications (Germany: Springer-Verlag, 1972).Google Scholar
Ma, Q., Wang, J. and Liu, T.. Time-dependent asymptotic behavior of the solution for wave equations with linear memory. Comput. Math. Appl. 76 (2018), 1372–1387.CrossRefGoogle Scholar
Ma, Q., Wang, X. and Xu, L.. Existence and regularity of time-dependent global attractors for the nonclassical reaction-diffusion equations with lower forcing term. Boundary Value Probl. 2016 (2016), 1–11.Google Scholar
Meng, F. and Liu, C.. Necessary and sufficient condition for the existence of time-dependent global attractor and application. J. Math. Phys. 58 (2017), 1–9.CrossRefGoogle Scholar
Pata, V. and Conti, M.. Asymptotic structure of the attractor for process on time-dependent spaces. Nonlinear Anal. RWA 19 (2014), 1–10.Google Scholar
Peng, X., Shang, Y. and Zheng, X.. Pullback attractors of nonautonomous nonclassical diffusion equations with nonlocal diffusion. Z. Angew. Math. Phys. 69 (2018), 1–14.CrossRefGoogle Scholar
Plinio, F. D., Duan, G. and Temam, R.. Time dependent attractor for the oscillon equation. Discrete Contin. Dyn. Syst. 29 (2011), 141–167.CrossRefGoogle Scholar
Qin, Y.. Integral and discrete inequalities and their applications, Vol I (Switzerland: Springer International Publishing AG, 2016).Google Scholar
Qin, Y.. Integral and discrete inequalities and their applications, Vol II (Switzerland: Springer International Publishing AG, 2016).Google Scholar
Qin, Y.. Analytic inequalities and their applications in PDEs (Switzerland: Birkhauser Verlag AG, 2017).CrossRefGoogle Scholar
Robinson, J. C.. Infinite-dimensional dynamical systems (England: Cambridge University Press, 2011).Google Scholar
Sun, C., Wang, S. and Zhong, C.. Global attractors for a nonclassical diffusion equation. Acta Math. Sin. Engl. Ser. 23 (2007), 1271–1280.CrossRefGoogle Scholar
Sun, C. and Yang, M.. Dynamics of the nonclassical diffusion equations. Asymptot. Anal. 59 (2008), 51–81.Google Scholar
Wang, J. and Ma, Q.. Asymptotic dynamic of the nonclassical diffusion equation with time-dependent coefficient. J. Appl. Anal. Comput. 11 (2021), 445–463.Google Scholar
Wang, Y., Zhu, Z. and Li, P.. Regularity of pullback attractors for non-autonomous nonclassical diffusion equations. J. Math. Anal. Appl. 459 (2018), 16–31.CrossRefGoogle Scholar
Zelik, S.. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Commun. Pure Appl. Anal. 3 (2004), 921–934.CrossRefGoogle Scholar
Zhu, K. and Sun, C.. Pullback attractors for nonclassical diffusion equations with delays. J. Math. Phys. 56 (2015), 1–20.CrossRefGoogle Scholar
Zhu, K., Xie, Y. and Zhou, F.. Attractors for the nonclassical reaction-diffusion equations on time-dependent spaces. Boundary Value Probl. (2020), 1–14. DOI: 10.1186/s13661-020-01392-7.Google Scholar