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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

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.
Citation: Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure and 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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[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.

[43]

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

[44]

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

[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.

[46]

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

[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.

[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.

[49]

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

[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.

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[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.

[43]

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

[44]

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

[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.

[46]

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

[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.

[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.

[49]

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

[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.

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

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