Acknowledgement
The authors would like to thank the reviewers for the helpful comments and suggestions which improved the presentation of the paper. This work was completed while the author was visiting the Vietnam Institute of Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for its hospitality.
References
- S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal. 20 (1999), no. 3-4, 263-277.
- G. Cantin, Non-existence of the global attractor for a partly dissipative reaction-diffusion system with hysteresis, J. Differential Equations 299 (2021), 333-361. https://doi.org/10.1016/j.jde.2021.07.023
- V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, 49, Amer. Math. Soc., Providence, RI, 2002.
- R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961), 445-466. https://doi.org/10.1016/S0006-3495(61)86902-6
- S. Gatti, A. Miranville, V. Pata, and S. Zelik, Attractors for semi-linear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math. 38 (2008), no. 4, 1117-1138. https://doi.org/10.1216/RMJ-2008-38-4-1117
- P. G. Geredeli and A. K. Khanmamedov, Long-time dynamics of the parabolic p-Laplacian equation, Commun. Pure Appl. Anal. 12 (2013), no. 2, 735-754. https://doi.org/10.3934/cpaa.2013.12.735
- C. Giorgi, V. Pata, and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 3, 333-354. https://doi.org/10.1007/s000300050049
- M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution equations, semigroups and functional analysis (Milano, 2000), 155-178, Progr. Nonlinear Differential Equations Appl., 50, Birkhauser, Basel, 2002.
- M. Krasnoselskii, P. Zabreiko, E. Pustylnik, and P. Sobolevskii, Integral operators in spaces of summable functions, translated from the Russian by T. Ando, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff, Leiden, 1976.
- J. Lee and Toi Vu Manh, Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity, Appl. Anal. 100 (2021), no. 4, 735-751. https://doi.org/10.1080/00036811.2019.1620214
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer, New York, 1972.
- Y. Lu and Z. D. Shao, Determining nodes for partly dissipative reaction diffusion systems, Nonlinear Anal. 54 (2003), no. 5, 873-884. https://doi.org/10.1016/S0362-546X(03)00112-3
- M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl. 143 (1989), no. 2, 295-326. https://doi.org/10.1016/0022-247X(89)90043-7
- M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal. 20 (1989), no. 4, 816-844. https://doi.org/10.1137/0520057
- J. Nagumo, S. Arimoto, and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E. 50 (1964), 2061-2070. https://doi.org/10.1109/JRPROC.1962.288235
- X. Pu and X. Zhang, A remark on partly dissipative reaction diffusion systems on ℝn, Acta Math. Appl. Sin. Engl. Ser. 24 (2008), no. 4, 583-588. https://doi.org/10.1007/s10255-005-5237-1
- J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge Univ. Press, Cambridge, 2001.
- A. Rodr'iguez-Bernal and B. X. Wang, Attractors for partly dissipative reaction diffusion systems in ℝn, J. Math. Anal. Appl. 252 (2000), no. 2, 790-803. https://doi.org/10.1006/jmaa.2000.7122
- Z. D. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions, J. Differential Equations 144 (1998), no. 1, 1-43. https://doi.org/10.1006/jdeq.1997.3383
- Z. D. Shao, Existence and continuity of strong solutions of partly dissipative reaction diffusion systems, Discrete Contin. Dyn. Syst. 2011 (2011), 1319-1328.
- W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), 543-551. http://projecteuclid.org/euclid.pjm/1102993723 102993723
- R. M. Temam, Navier-Stokes equations and nonlinear functional analysis, second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995. https://doi.org/10.1137/1.9781611970050
- L. T. Thuy and N. D. Toan, Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces, Math. Methods Appl. Sci. 44 (2021), no. 1, 820-852. https://doi.org/10.1002/mma.6791
- M. Yang and C. Zhong, The existence and uniqueness of the solutions for partly dissipative reaction diffusion systems in ℝn, J. Lanzhou Univ. Nat. Sci. 42 (2006), no. 5, 130-136.