# GLOBAL ATTRACTOR FOR COUPLED TWO-COMPARTMENT GRAY-SCOTT EQUATIONS

• Zhao, Xiaopeng ;
• Liu, Bo
• Published : 2013.01.31
• 40 5

#### Abstract

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

#### Keywords

global attractor;two-compartment Gray-Scott equations;regularity estimates

#### References

1. M. Ashkenazi and H. G. Othmer, Spatial patterns in coupled biochemical oscillators, J. Math. Biol. 5 (1978), no. 4, 305-350.
2. J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations I: evolution equations and the steady state solutions, Bull. Math. Biol. 37 (1975), no. 4, 323-365.
3. T. Dlotko, Global attractor for the Cahn-Hilliard equation in $H^2$ and $H^3$, J. Differential Equations 113 (1994), no. 2, 381-393. https://doi.org/10.1006/jdeq.1994.1129
4. P. Gormley, K. Li, and G. W. Irwin, Modeling molecular interaction pathways using a two-stage identification algorithm, Syst. Synth. Biol. 1 (2007), no. 3, 145-160. https://doi.org/10.1007/s11693-008-9012-5
5. P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), 29-43. https://doi.org/10.1016/0009-2509(83)80132-8
6. P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B ${\rightarrow}$ 3B, B ${\rightarrow}$ C, Chem. Eng. Sci. 39 (1984), 1087-1097. https://doi.org/10.1016/0009-2509(84)87017-7
7. P. Gray and S. K. Scott, Chemical Waves and Instabilities, Clarendon, Oxford, 1990.
8. J. K. Hale, L. A. Peletier, andW. C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000), no. 1, 102-130. https://doi.org/10.1137/S0036139998334913
9. M. Kawato and R. Suzuki, Two coupled neural oscillators as a model of the circadian pacemaker, J. Theoret. Biol. 86 (1980), no. 3, 547-575. https://doi.org/10.1016/0022-5193(80)90352-5
10. K. J. Lee, W. D. McCormick, J. E. Pearson, and H. L. Swinney, Experimental observations of self-replicating spots in a reaction-diffusion system, Nature 369 (1994), 215-218. https://doi.org/10.1038/369215a0
11. D. S. Li and C. K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations 149 (1998), no. 2, 191-210. https://doi.org/10.1006/jdeq.1998.3429
12. T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, Science Press, Beijing, 2006, (in Chinese).
13. J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal. Real World Appl. 5 (2004), no. 1, 105-121. https://doi.org/10.1016/S1468-1218(03)00020-8
14. Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Phys. D 150 (2001), 137-162. https://doi.org/10.1016/S0167-2789(00)00214-1
15. I. Schreiber and M. Marek, Strange attractors in coupled reaction-diffusion cells, Phys. D 5 (1982), no. 2-3, 258-272. https://doi.org/10.1016/0167-2789(82)90021-5
16. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.
17. L. Song, Y. Zhang, and T. Ma, Global attractor of the Cahn-Hilliard euqation in $H^k$ spaces, J. Math. Anal. Appl. 355 (2009), no. 1, 53-62. https://doi.org/10.1016/j.jmaa.2009.01.035
18. L. Song, Y. Zhang, and T. Ma, Global attractor of a modified Swift-Hohenberg equation in \$H^k space, Nonlinear Anal. 72 (2010), no. 1, 183-191. https://doi.org/10.1016/j.na.2009.06.103
19. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
20. Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Anal. 74 (2011), no. 5, 1969-1986. https://doi.org/10.1016/j.na.2010.11.004
21. Y. You, Global dynamics of a reaction-diffusion system, Electron. J. Differential Equations 2011 (2011), no. 25, 1-28.
22. X. Zhao and B. Liu, The existence of global attractor for convective Cahn-Hilliard equation, J. Korean Math. Soc. 49 (2012), no. 2, 357-378. https://doi.org/10.4134/JKMS.2012.49.2.357
23. X. Zhao and C. Liu, The existence of global attractor for a fourth-order parabolic equation, Appl. Anal. 92 (2013), no. 1, 44-59. https://doi.org/10.1080/00036811.2011.590476
24. S. Zheng and A. Milani, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Differential Equations 209 (2005), no. 1, 101-139. https://doi.org/10.1016/j.jde.2004.08.026

#### Acknowledgement

Supported by : Jilin University