# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022103
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## Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2 Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China

*Corresponding author: Zhong-Xin Ma

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: Both of the authors are supported by the NSFC of China (No. 11971317)

In this paper, we consider a partly dissipative reaction-diffusion system with discontinuous nonlinearity in the form
 $\begin{equation*} \left\{\begin{array}{ll} u_t-\Delta u+u+w\in H_0(u-a), \\ w_t-\epsilon(u-\gamma w) = 0, \end{array}\right. \end{equation*}$
where
 $H_0$
is a multi-valued function of Heaviside type. This type of system is used for describing the generation and transmission of electrical signals in neuroscience. We first present an existence result on global solutions. Then, we prove that the system possesses a global attractor having the
 $H^r\times H^r$
-regularity
 $(0\leq r<2)$
. Moreover, by showing the Kneser property for the system, the global attractor is proved to be connected. The main characteristic of the system is that the linear part cannot be represented as the subdifferential of a compact-type function.
Citation: Jia-Cheng Zhao, Zhong-Xin Ma. Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022103
##### References:
 [1] B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan, Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3787-3797.  doi: 10.3934/dcdsb.2018077. [2] J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. [4] D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044. [5] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971,101–156. doi: 10.1016/B978-0-12-775850-3.50009-1. [6] R. Caballero, A. N. Carvalho, P. Marín-Rubio and J. Valero, Robustness of dynamically gradient multivalued dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1049-1077.  doi: 10.3934/dcdsb.2019006. [7] T. Caraballo, J. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Rev. Mat. Complut., 33 (2020), 583-617.  doi: 10.1007/s13163-019-00323-0. [8] K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, 1, Walter de Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228. [9] P. Fabrie and C. Galusinski, Exponential attractors for a partially dissipative reaction system, Asymptotic Anal., 12 (1996), 329-354.  doi: 10.3233/ASY-1996-12403. [10] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., 17 (1955), 257-278.  doi: 10.1007/BF02477753. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. [12] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.  doi: 10.1113/jphysiol.1952.sp004764. [13] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. 1. Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. [14] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040. [15] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155. [16] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264.  doi: 10.12785/amis. [17] O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinski, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations Without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, 2008. [18] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. [19] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw 1968. [20] J. Lee and V. M. Toi, Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity, Appl. Anal., 100 (2021), 735-751.  doi: 10.1080/00036811.2019.1620214. [21] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057. [22] V. S. Mel'nik, Families of multivalued semiflows and their attractors, Dokl. Akad. Nauk., 353 (1997), 158-162. [23] V. S. Mel'nik, Multivalued semiflows and their attractors, Dokl. Akad. Nauk., 343 (1995), 302-305. [24] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235. [25] M. Ôtani, On the existence of strong solutions for $\frac {du}{dt}(t)+\partial\psi^1(u(t))-\partial\psi^2(u(t))\ni f(t)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 575-605. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X. [28] J. Valero, Attractors of parabolic equations without uniqueness, J. Dynam. Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800. [29] J. Valero, Characterization of the attractor for nonautonomous reaction-diffusion equations with discontinuous nonlinearity, J. Differential Equations, 275 (2021), 270-308.  doi: 10.1016/j.jde.2020.11.036. [30] J. Valero, On the Kneser property for some parabolic problems, Topology Appl., 153 (2005), 975-989.  doi: 10.1016/j.topol.2005.01.025. [31] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. [32] R.-N. Wang, Z.-X. Ma and A. Miranville, Topological structure of the solution sets for a nonlinear delay evolution, Int. Math. Res. Not. IMRN, 2022 (2022), 4801-4889.  doi: 10.1093/imrn/rnab176.

show all references

##### References:
 [1] B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan, Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3787-3797.  doi: 10.3934/dcdsb.2018077. [2] J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. [4] D. Bothe, Multivalued perturbations of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044. [5] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971,101–156. doi: 10.1016/B978-0-12-775850-3.50009-1. [6] R. Caballero, A. N. Carvalho, P. Marín-Rubio and J. Valero, Robustness of dynamically gradient multivalued dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1049-1077.  doi: 10.3934/dcdsb.2019006. [7] T. Caraballo, J. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Rev. Mat. Complut., 33 (2020), 583-617.  doi: 10.1007/s13163-019-00323-0. [8] K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, 1, Walter de Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228. [9] P. Fabrie and C. Galusinski, Exponential attractors for a partially dissipative reaction system, Asymptotic Anal., 12 (1996), 329-354.  doi: 10.3233/ASY-1996-12403. [10] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., 17 (1955), 257-278.  doi: 10.1007/BF02477753. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647. [12] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.  doi: 10.1113/jphysiol.1952.sp004764. [13] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. 1. Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, 1997. [14] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040. [15] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155. [16] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264.  doi: 10.12785/amis. [17] O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinski, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations Without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, 2008. [18] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. [19] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw 1968. [20] J. Lee and V. M. Toi, Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity, Appl. Anal., 100 (2021), 735-751.  doi: 10.1080/00036811.2019.1620214. [21] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057. [22] V. S. Mel'nik, Families of multivalued semiflows and their attractors, Dokl. Akad. Nauk., 353 (1997), 158-162. [23] V. S. Mel'nik, Multivalued semiflows and their attractors, Dokl. Akad. Nauk., 343 (1995), 302-305. [24] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235. [25] M. Ôtani, On the existence of strong solutions for $\frac {du}{dt}(t)+\partial\psi^1(u(t))-\partial\psi^2(u(t))\ni f(t)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 575-605. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X. [28] J. Valero, Attractors of parabolic equations without uniqueness, J. Dynam. Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800. [29] J. Valero, Characterization of the attractor for nonautonomous reaction-diffusion equations with discontinuous nonlinearity, J. Differential Equations, 275 (2021), 270-308.  doi: 10.1016/j.jde.2020.11.036. [30] J. Valero, On the Kneser property for some parabolic problems, Topology Appl., 153 (2005), 975-989.  doi: 10.1016/j.topol.2005.01.025. [31] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. [32] R.-N. Wang, Z.-X. Ma and A. Miranville, Topological structure of the solution sets for a nonlinear delay evolution, Int. Math. Res. Not. IMRN, 2022 (2022), 4801-4889.  doi: 10.1093/imrn/rnab176.
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