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September  2011, 10(5): 1415-1445. doi: 10.3934/cpaa.2011.10.1415

Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems

Yuncheng You 1,

1. 

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620

Received  May 2009 Revised  March 2010 Published  April 2011

In this paper we prove the existence of a global attractor, an $(H,E)$ global attractor, and an exponential attractor for the cubic autocatalytic reaction-diffusion systems represented by the reversible Gray-Scott equations. The two pairs of oppositely signed nonlinear terms feature the challenge in conducting various estimates. A new rescaling and grouping estimation method is introduced and combined with the other approaches to achieve the proof of dissipation, asymptotic compactness, and discrete squeezing property in all the stages.
Keywords: cubic autocatalysis, Reversible reaction-diffusion system, Gray-Scott equations, exponential attractor., global attractor, asymptotic dynamics.
Mathematics Subject Classification: Primary: 37L30; Secondary: 35B40, 35B41, 35K55, 35K57, 35Q8.
Citation: Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415
References:
[1]

J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations. I. Evolution equations and the steady state solutions, Bull. Math. Biology, 37 (1975), 323-365.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolutionary Equations," Nauka, Moskow, 1989; English translation, North-Holland, Amsterdam, 1992.  Google Scholar

[4]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Diff. Eqns., 7 (1995), 567-590.  Google Scholar

[5]

D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biology, 55 (1993), 365-384. Google Scholar

[6]

K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Analysis, 24 (1995), 1713-1725.  Google Scholar

[7]

V. V. Chepyzhov and M.I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar

[8]

E. J. Crampin and P. K. Maini, Reaction-diffusion models for biological pattern formation, Methods Appl. Anal., 8 (2001), 415-428.  Google Scholar

[9]

A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  Google Scholar

[10]

L. Dung, Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension, in "Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations," Wilmington, NC, USA, (2002), 536-543.  Google Scholar

[11]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dynamics and Diff. Eqns., 13 (2001), 791-806.  Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John Wiley & Sons, Chichester; Masson, Paris, 1994.  Google Scholar

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction diffusion systems in $\mathbbR^3$, C.R. Acad. Sci., 330 (2000), 713-718.  Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  Google Scholar

[15]

I. R. Epstein, Complex dynamical behavior in simple chemical systems, J. Phys. Chemistry, 88 (1984), 187-198. Google Scholar

[16]

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. Google Scholar

[17]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $a+2b\to 3b,b\to c$, Chem. Eng. Sci., 39 (1984), 1087-1097. Google Scholar

[18]

P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities," Clarendon, Oxford, 1994. Google Scholar

[19]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, Berlin, 1981.  Google Scholar

[20]

T. Kolokolnikov, T. Erneux, and J. Wei, Mesa-type patterns in one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77.  Google Scholar

[21]

T. Kolokolnikov and J. Wei, On ring-like solutions for the Gray-Scott model: Existence, instability and self-replicating rings, Euro. J. Appl. Math., 16 (2005), 201-237.  Google Scholar

[22]

K. J. Lee, W. D. McCormick, Q. Ouyang and H. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. Google Scholar

[23]

K. J. Lee, W. D. McCormick, J. E. Pearson and H. L. Swinney, Experimental observation of self-replicating spots in areaction-diffusion system, Nature, 369 (1994), 215-218. Google Scholar

[24]

K. J. Lee and H. L. Swinney, Replicating spots in reaction-diffusion systems, Int. J. Bifurcation and Chaos, 7 (1997), 1149-1158. Google Scholar

[25]

K. Matsuura and M. Ôtani, Exponential attarctors for a quasilinear parabolic equation, Disc. Cont. Dyn. Sys. Suppl., (2007), 713-720.  Google Scholar

[26]

J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Analysis: Real World Applications, \tbf{5} (2004), 105-121.  Google Scholar

[27]

A. J. Milani and N. J. Koksch, "An Introduction to Semiflows," Chapman & Hall/CRC, 2005.  Google Scholar

[28]

D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots, Physica D, 192 (2004), 33-62.  Google Scholar

[29]

C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model, J. Phys. A, 33 (2000), 8893-8916.  Google Scholar

[30]

K. Osaki, T. Tsujikawa, A. Yagi and M. Minura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.  Google Scholar

[31]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. Google Scholar

[32]

I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 48 (1968), 1695-1700. Google Scholar

[33]

W. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems, Phys. Rev. E, 56 (1997), 185-198.  Google Scholar

[34]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge University Press, Cambridge, UK, 2001.  Google Scholar

[35]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Lecture Notes in Math, 1072, Springer-Verlag, Berlin, 1984.  Google Scholar

[36]

J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biology, 81 (1979), 389-400. Google Scholar

[37]

S. K. Scott and K. Showalter, Simple and complex reaction-diffusion fronts, in "Chemical Waves and Patterns" (R. Kapral and K. Showalter, eds.), Springer, 10 (1995), 485-516. Google Scholar

[38]

E. E. Selkov, Self-oscillations in glycolysis: a simple kinetic model, European J. Biochem., 4 (1968), 79-86. Google Scholar

[39]

George R. Sell and Yuncheng You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002.  Google Scholar

[40]

L. J. Shaw and J. D. Murray, Analysis of a model for complex skin patterns, SIAM J. Appl. Math., 50 (1990), 628-648.  Google Scholar

[41]

M. Stanislavova, A. Stefanov and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahonay equations on $\Re^3$, J. Dff. Eqns., 219 (2005), 451-483.  Google Scholar

[42]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distributed derivatives in unbounded domains, Nonlinear Analysis, 63 (2005), 49-65.  Google Scholar

[43]

Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988.  Google Scholar

[44]

B. Wang, Attractors for reaction-diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52.  Google Scholar

[45]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model, Stud. Appl. Math., 109 (2002), 229-264.  Google Scholar

[46]

J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\Re^2$, Stud. Appl. Math., 110 (2003), 63-102.  Google Scholar

[47]

J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $\Re^2$, Physica D, 176 (2003), 147-180.  Google Scholar

[48]

L. Yang, A. M. Zhabotinsky and I. R. Epstein, Stable square and other oscillatory Turing patterns in a reaction-diffusion model, Phys. Rev. Lett., 92 (2004), 198303-1-4. Google Scholar

[49]

Y. You, Global dynamics of nonlinear wave equations with cubic non-monotone damping, Dynamics of PDE, 1 (2004), 65-86.  Google Scholar

[50]

Y. You, Finite dimensional reduction of global dynamics and lattice dynamics of a damped nonlinear wave equation, in "Control Theory and mathematical Finance" (S. Tang and J. Yong, eds.), World Scientific, (2007), 367-386.  Google Scholar

[51]

Y. You, Global dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.  Google Scholar

[52]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970.  Google Scholar

[53]

Y. You, Inertial manifolds for nonautonomous skew product semiflows, Far East J. Appl. Math., 32 (2008), 141-188.  Google Scholar

[54]

Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Systems, Series S, 2 (2009), 193-219.  Google Scholar

[55]

Y. You, Asymptotic dynamics of the modified Schnackenberg equations, Disc. Cont. Dyn. Systems, Supplement 2009, 857-868.  Google Scholar

show all references

References:
[1]

J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations. I. Evolution equations and the steady state solutions, Bull. Math. Biology, 37 (1975), 323-365.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolutionary Equations," Nauka, Moskow, 1989; English translation, North-Holland, Amsterdam, 1992.  Google Scholar

[4]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Diff. Eqns., 7 (1995), 567-590.  Google Scholar

[5]

D. L. Benson, J. A. Sherratt and P. K. Maini, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biology, 55 (1993), 365-384. Google Scholar

[6]

K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Analysis, 24 (1995), 1713-1725.  Google Scholar

[7]

V. V. Chepyzhov and M.I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002.  Google Scholar

[8]

E. J. Crampin and P. K. Maini, Reaction-diffusion models for biological pattern formation, Methods Appl. Anal., 8 (2001), 415-428.  Google Scholar

[9]

A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  Google Scholar

[10]

L. Dung, Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension, in "Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations," Wilmington, NC, USA, (2002), 536-543.  Google Scholar

[11]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dynamics and Diff. Eqns., 13 (2001), 791-806.  Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," John Wiley & Sons, Chichester; Masson, Paris, 1994.  Google Scholar

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction diffusion systems in $\mathbbR^3$, C.R. Acad. Sci., 330 (2000), 713-718.  Google Scholar

[14]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  Google Scholar

[15]

I. R. Epstein, Complex dynamical behavior in simple chemical systems, J. Phys. Chemistry, 88 (1984), 187-198. Google Scholar

[16]

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. Google Scholar

[17]

P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $a+2b\to 3b,b\to c$, Chem. Eng. Sci., 39 (1984), 1087-1097. Google Scholar

[18]

P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities," Clarendon, Oxford, 1994. Google Scholar

[19]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, Berlin, 1981.  Google Scholar

[20]

T. Kolokolnikov, T. Erneux, and J. Wei, Mesa-type patterns in one-dimensional Brusselator and their stability, Physica D, 214 (2006), 63-77.  Google Scholar

[21]

T. Kolokolnikov and J. Wei, On ring-like solutions for the Gray-Scott model: Existence, instability and self-replicating rings, Euro. J. Appl. Math., 16 (2005), 201-237.  Google Scholar

[22]

K. J. Lee, W. D. McCormick, Q. Ouyang and H. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194. Google Scholar

[23]

K. J. Lee, W. D. McCormick, J. E. Pearson and H. L. Swinney, Experimental observation of self-replicating spots in areaction-diffusion system, Nature, 369 (1994), 215-218. Google Scholar

[24]

K. J. Lee and H. L. Swinney, Replicating spots in reaction-diffusion systems, Int. J. Bifurcation and Chaos, 7 (1997), 1149-1158. Google Scholar

[25]

K. Matsuura and M. Ôtani, Exponential attarctors for a quasilinear parabolic equation, Disc. Cont. Dyn. Sys. Suppl., (2007), 713-720.  Google Scholar

[26]

J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Analysis: Real World Applications, \tbf{5} (2004), 105-121.  Google Scholar

[27]

A. J. Milani and N. J. Koksch, "An Introduction to Semiflows," Chapman & Hall/CRC, 2005.  Google Scholar

[28]

D. Morgan and T. Kaper, Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots, Physica D, 192 (2004), 33-62.  Google Scholar

[29]

C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model, J. Phys. A, 33 (2000), 8893-8916.  Google Scholar

[30]

K. Osaki, T. Tsujikawa, A. Yagi and M. Minura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.  Google Scholar

[31]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. Google Scholar

[32]

I. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems, J. Chem. Physics, 48 (1968), 1695-1700. Google Scholar

[33]

W. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems, Phys. Rev. E, 56 (1997), 185-198.  Google Scholar

[34]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge University Press, Cambridge, UK, 2001.  Google Scholar

[35]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Lecture Notes in Math, 1072, Springer-Verlag, Berlin, 1984.  Google Scholar

[36]

J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biology, 81 (1979), 389-400. Google Scholar

[37]

S. K. Scott and K. Showalter, Simple and complex reaction-diffusion fronts, in "Chemical Waves and Patterns" (R. Kapral and K. Showalter, eds.), Springer, 10 (1995), 485-516. Google Scholar

[38]

E. E. Selkov, Self-oscillations in glycolysis: a simple kinetic model, European J. Biochem., 4 (1968), 79-86. Google Scholar

[39]

George R. Sell and Yuncheng You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002.  Google Scholar

[40]

L. J. Shaw and J. D. Murray, Analysis of a model for complex skin patterns, SIAM J. Appl. Math., 50 (1990), 628-648.  Google Scholar

[41]

M. Stanislavova, A. Stefanov and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahonay equations on $\Re^3$, J. Dff. Eqns., 219 (2005), 451-483.  Google Scholar

[42]

C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distributed derivatives in unbounded domains, Nonlinear Analysis, 63 (2005), 49-65.  Google Scholar

[43]

Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988.  Google Scholar

[44]

B. Wang, Attractors for reaction-diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52.  Google Scholar

[45]

M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnackenberg model, Stud. Appl. Math., 109 (2002), 229-264.  Google Scholar

[46]

J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\Re^2$, Stud. Appl. Math., 110 (2003), 63-102.  Google Scholar

[47]

J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $\Re^2$, Physica D, 176 (2003), 147-180.  Google Scholar

[48]

L. Yang, A. M. Zhabotinsky and I. R. Epstein, Stable square and other oscillatory Turing patterns in a reaction-diffusion model, Phys. Rev. Lett., 92 (2004), 198303-1-4. Google Scholar

[49]

Y. You, Global dynamics of nonlinear wave equations with cubic non-monotone damping, Dynamics of PDE, 1 (2004), 65-86.  Google Scholar

[50]

Y. You, Finite dimensional reduction of global dynamics and lattice dynamics of a damped nonlinear wave equation, in "Control Theory and mathematical Finance" (S. Tang and J. Yong, eds.), World Scientific, (2007), 367-386.  Google Scholar

[51]

Y. You, Global dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.  Google Scholar

[52]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure Appl. Anal., 7 (2008), 947-970.  Google Scholar

[53]

Y. You, Inertial manifolds for nonautonomous skew product semiflows, Far East J. Appl. Math., 32 (2008), 141-188.  Google Scholar

[54]

Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Systems, Series S, 2 (2009), 193-219.  Google Scholar

[55]

Y. You, Asymptotic dynamics of the modified Schnackenberg equations, Disc. Cont. Dyn. Systems, Supplement 2009, 857-868.  Google Scholar

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