# American Institute of Mathematical Sciences

August  2020, 25(8): 3135-3152. doi: 10.3934/dcdsb.2020054

## Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions

 1 Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Korea 2 Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam

* Corresponding author: Vu Manh Toi

Received  May 2019 Revised  October 2019 Published  August 2020 Early access  February 2020

Fund Project: The first author is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2019R1A6A3A01091340). The second author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2018.303

In this paper we study the asymptotic behavior of solutions for a class of nonautonomous reaction-diffusion equations with dynamic boundary conditions possessing finite delay. Under the polynomial conditions of reaction term, suitable conditions of delay terms and a minimal conditions of time-dependent force functions, we first prove the existence and uniqueness of solutions by using the Galerkin method. Then, we ensure the existence of pullback attractors for the associated process to the problem by proving some uniform estimates and asymptotic compactness properties (via an energy method). With an additional condition of time-dependent force functions, we prove that the boundedness of pullback attractors in smoother spaces.

Citation: Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054
##### References:
 [1] M. Anguiano, P. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046. [2] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  doi: 10.1080/03605309308820976. [3] Z. H. Fan and C. K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005. [4] K. Fellner, S. Sonner, B. Q. Tang and D. D. Thuan, Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4055-4078.  doi: 10.3934/dcdsb.2019050. [5] J. García-Luengo and P. Marín-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. [6] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010. [7] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition, J. Nonlinear Sci., 22 (2012), 85-106.  doi: 10.1007/s00332-011-9109-y. [8] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [9] J. K. Hale, Theory of Functional Differential Equations. Applied Mathematical Sciences, vol. 3. Springer, Berlin, 1977. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] H. Harraga and M. Yebdri, Attractors for a nonautonomous reaction-diffusion equation with delay, Appl. Math. Nonlinear Sci., 3 (2018), 127-150.  doi: 10.21042/AMNS.2018.1.00010. [12] T. D. Ke and N. C. Wong, Asymptotic behavior for retarded parabolic equations with superlinear perturbations, J. Optim. Theory Appl., 146 (2010), 117-135.  doi: 10.1007/s10957-010-9665-6. [13] V. B. Kolmanovskii and A. D. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. [14] R. Samprogna and T. Caraballo, Pullback attractor for a dynamic boundary non-autonomous problem with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 509-523.  doi: 10.3934/dcdsb.2017195. [15] J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012. [16] L. Yang, Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition, Nonlinear Anal., 71 (2009), 4012-4025.  doi: 10.1016/j.na.2009.02.083. [17] L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022. [18] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.

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##### References:
 [1] M. Anguiano, P. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046. [2] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  doi: 10.1080/03605309308820976. [3] Z. H. Fan and C. K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005. [4] K. Fellner, S. Sonner, B. Q. Tang and D. D. Thuan, Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4055-4078.  doi: 10.3934/dcdsb.2019050. [5] J. García-Luengo and P. Marín-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. [6] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010. [7] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition, J. Nonlinear Sci., 22 (2012), 85-106.  doi: 10.1007/s00332-011-9109-y. [8] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [9] J. K. Hale, Theory of Functional Differential Equations. Applied Mathematical Sciences, vol. 3. Springer, Berlin, 1977. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] H. Harraga and M. Yebdri, Attractors for a nonautonomous reaction-diffusion equation with delay, Appl. Math. Nonlinear Sci., 3 (2018), 127-150.  doi: 10.21042/AMNS.2018.1.00010. [12] T. D. Ke and N. C. Wong, Asymptotic behavior for retarded parabolic equations with superlinear perturbations, J. Optim. Theory Appl., 146 (2010), 117-135.  doi: 10.1007/s10957-010-9665-6. [13] V. B. Kolmanovskii and A. D. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. [14] R. Samprogna and T. Caraballo, Pullback attractor for a dynamic boundary non-autonomous problem with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 509-523.  doi: 10.3934/dcdsb.2017195. [15] J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012. [16] L. Yang, Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition, Nonlinear Anal., 71 (2009), 4012-4025.  doi: 10.1016/j.na.2009.02.083. [17] L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022. [18] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.
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