# American Institue of Mathematical Sciences

2017, 22(10): 3691-3706. doi: 10.3934/dcdsb.2017150

## Pullback attractors of FitzHugh-Nagumo system on the time-varying domains

 1 College of Science, National University of Defense Technology, Changsha 410073, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

* Corresponding author: Jianhua Huang

Received  December 2016 Revised  January 2017 Published  April 2017

Fund Project: The authors are supported by the NSF of China(No.11371367,11571126), the third author is also supported by the Fundamental Research Funds for the Central Universities (106112016CDJZR105501)

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

Citation: Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
##### References:
 [1] S. Bonaccorsi, G. Guatteri, A variational approach to evolution problems with variable domains, J. Differential Equations, 175 (2001), 51-70. doi: 10.1006/jdeq.2000.3959. [2] H. Crauel, P. Kloeden, M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292. [3] L. Evans, Partial Differential Equations, Grad. Stud. Math. , Amer. Math. Soc. , 19 Providence, RI, 1998. [4] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1861), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [5] C. He, L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291. doi: 10.1006/jdeq.1999.3702. [6] P. Kloeden, José Real, C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential equations, 246 (2009), 4702-4730. doi: 10.1016/j.jde.2008.11.017. [7] P. Kloeden, P. Maín-Rubio, J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090. doi: 10.1016/j.jde.2007.10.031. [8] J. Límaco, L. A. Medeiros, E. Zuazua, Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains, Mat. Contemp., 23 (2002), 49-70. [9] J. Lions, Quelques méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod, Paris, 1969. [10] W. Liu, B. Wang, Asymptotic behavior of the FitzHugh-Nagumo system, Inter. J. Evolution Equations, 2 (2007), 129-163. [11] Y. Lu, Z. Shao, Determining nodes for partly dissipative reaction systems, Nonlinear Anal, 54 (2003), 873-884. doi: 10.1016/S0362-546X(03)00112-3. [12] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057. [13] M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326. doi: 10.1016/0022-247X(89)90043-7. [14] J. Nagumo, S. Arimoto, S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070. doi: 10.1109/JRPROC.1962.288235. [15] J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. [16] C. Sun, Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177. [17] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [18] B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.

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##### References:
 [1] S. Bonaccorsi, G. Guatteri, A variational approach to evolution problems with variable domains, J. Differential Equations, 175 (2001), 51-70. doi: 10.1006/jdeq.2000.3959. [2] H. Crauel, P. Kloeden, M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292. [3] L. Evans, Partial Differential Equations, Grad. Stud. Math. , Amer. Math. Soc. , 19 Providence, RI, 1998. [4] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1861), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [5] C. He, L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291. doi: 10.1006/jdeq.1999.3702. [6] P. Kloeden, José Real, C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential equations, 246 (2009), 4702-4730. doi: 10.1016/j.jde.2008.11.017. [7] P. Kloeden, P. Maín-Rubio, J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090. doi: 10.1016/j.jde.2007.10.031. [8] J. Límaco, L. A. Medeiros, E. Zuazua, Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains, Mat. Contemp., 23 (2002), 49-70. [9] J. Lions, Quelques méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod, Paris, 1969. [10] W. Liu, B. Wang, Asymptotic behavior of the FitzHugh-Nagumo system, Inter. J. Evolution Equations, 2 (2007), 129-163. [11] Y. Lu, Z. Shao, Determining nodes for partly dissipative reaction systems, Nonlinear Anal, 54 (2003), 873-884. doi: 10.1016/S0362-546X(03)00112-3. [12] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057. [13] M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326. doi: 10.1016/0022-247X(89)90043-7. [14] J. Nagumo, S. Arimoto, S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070. doi: 10.1109/JRPROC.1962.288235. [15] J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. [16] C. Sun, Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177. [17] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [18] B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.
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