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June  2017, 22(4): 1601-1633. doi: 10.3934/dcdsb.2017033

Stability of traveling waves for autocatalytic reaction systems with strong decay

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author

Received  May 2016 Revised  June 2016 Published  November 2016

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system $u_{t}=d u_{xx}-uv$, $v_{t}=v_{xx}+uv-Kv^{q}$ with $q>1$. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

Citation: Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033
References:
[1]

A. L. Afendikov and T. J. Bridges, Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 257-272. doi: 10.1098/rspa.2000.0665.

[2]

J. AlexanderR. Gardner and C. Jones, A topological invariant arising in the stability of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212.

[3]

J. C. Alexander and R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World, 2 (1995), 471-507.

[4]

L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232. doi: 10.1007/s002110100365.

[5]

B. Barker, J. Humpherys and K. Zumbrun, STABLAB: A MATLAB-based Numerical Library for Evans Function Computation Version 1. 0 preprint. Available from: http://impact.byu.edu/stablab/STABLAB_1.0_doc.pdf.

[6]

J. Billingham and D. J. Needham, A note on the properties of a family of traveling wave solutions arising in cubic autocatalysis, Dynam. Stability Systems, 6 (1991), 33-49. doi: 10.1080/02681119108806105.

[7]

T. J. BridgesG. Derks and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Phys. D, 172 (2002), 190-216. doi: 10.1016/S0167-2789(02)00655-3.

[8]

J. Carr, Application Of Centre Manifold Theory Springer-Verlag, New York-Berlin, 1981.

[9]

X. ChenY. Qi and Y. Zhang, Existence of traveling waves of auto-catalytic systems with decay, J. Differential Equations, 260 (2016), 7982-7999. doi: 10.1016/j.jde.2016.02.009.

[10]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations D. C. Heath and Co. , Boston, Mass, 1965.

[11]

S. Focant and Th. Galley, Existence and stability of propagating fronts for an autocatalytic reactio-diffuion systems, Physica D, 120 (1998), 346-368. doi: 10.1016/S0167-2789(98)00096-7.

[12]

S.-C. Fu and J.-C. Tsai, The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364. doi: 10.1016/j.jde.2014.02.009.

[13]

P. GrayJ. F. Griffiths and S. K. Scott, Oscillations, Glow and Ignition in Carbon Monoxide Oxidation in an Open System Ⅰ. Experimental Studies of the Ignition Diagram and the Effects of Added Hydrogen, Proc. R. Soc. Lond. A, 397 (1985), 21-44. doi: 10.1098/rspa.1985.0002.

[14]

P. Gray, Instabilities and oscillations in chemical reactions in closed and open systems, Proc. R. Soc. Lond. A, 415 (1988), 1-34.

[15]

A. HannaA. Saul and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Am. Chem. Soc., 104 (1982), 3838-3844. doi: 10.1021/ja00378a011.

[16]

D. Henry, Geometric Theory Of Semilinear Parabolic Equations Springer-Verlag, Berlin-New York, 1981.

[17]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59. doi: 10.1016/j.jmaa.2005.04.084.

[18]

P. Howard and K. Zumbrun, The Evans functions and stbility criteria for degenerate viscous shcok waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855. doi: 10.3934/dcds.2004.10.837.

[19]

E. Jakab, D. Horvath, J. H. Merkin, S. K. Scott, P. L. Simon and A. Toth, Isothermal flame balls: Effect of autocatalyst decay Phys. Rev. E 66 (2002), 016207. doi: 10.1103/PhysRevE.66.016207.

[20]

Y. Li and Y. Wu, Stability of traveling waves with non-critical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170. doi: 10.3934/dcdsb.2008.10.149.

[21]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521. doi: 10.1137/100814974.

[22]

Y. Li and Y. Wu, Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132. doi: 10.1051/mmnp/20138308.

[23]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.I. Permanent form travelling waves, SIAM J. Appl. Math., 59 (1999), 870-899. doi: 10.1137/S0036139996312594.

[24]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.Ⅱ. The initial boundary value problem, SIAM J. Appl. Math., 60 (2000), 1707-1748. doi: 10.1137/S0036139998344775.

[25]

J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system.Ⅳ. Quadratic autocatalysis with quadratic decay, Proc. R. Soc. Lond. A, 434 (1991), 531-554.

[26]

J. H. MerkinD. J. Needham and S. K. Scott, A simple model for sustained oscillations in isothermal branch-chain or autocatalytic reactions in a well stirred open system. Ⅰ. Stationary states and local stabilities, Proc. Roy. Soc. London Ser. A, 398 (1985), 81-100. doi: 10.1098/rspa.1985.0026.

[27]

J. H. MerkinD. J. Needham and S. K. Scott, The development of travelling waves in a simple isothermal chemical system.I. Quadratic autocatalysis with linear decay, Proc. Roy. Soc. London Ser. A, 424 (1989), 187-209. doi: 10.1098/rspa.1989.0075.

[28]

D. J. Needham, A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay, Z. Angew. Math. Phys., 42 (1991), 455-459. doi: 10.1007/BF00945715.

[29]

D. J. Needham and J. H. Merkin, The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Philos. Trans. Roy. Soc. London Ser. A, 337 (1991), 261-274. doi: 10.1098/rsta.1991.0122.

[30]

B. S. Ng and W. H. Reid, The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.

[31]

A. Pazy, Semigroups Of Linear Operators And Applications To Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[32]

B. P. Palka, An Introduction To Complex Function Theory Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0975-1.

[33]

R. L. PegoP. Smereka and M. I. Weinstein, Oscillatory instability of traveling waves for a KdV-Burgers equation, Phys. D, 67 (1993), 45-65. doi: 10.1016/0167-2789(93)90197-9.

[34]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instability of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055.

[35]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing. Second Edition Cambridge University Press, Cambridge, 1992.

[36]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs With MATLAB Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615542.

[37]

Y. WuX. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47.

[38]

Q. Ye, Z. Y. Li, M. X. Wang and Y. Wu, Introduction To Reaction Diffusion Equation. Second Edition Science Press, Beijing, 2011.

show all references

References:
[1]

A. L. Afendikov and T. J. Bridges, Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 257-272. doi: 10.1098/rspa.2000.0665.

[2]

J. AlexanderR. Gardner and C. Jones, A topological invariant arising in the stability of travelling waves, J. Reine Angew. Math., 410 (1990), 167-212.

[3]

J. C. Alexander and R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation, Nonlinear World, 2 (1995), 471-507.

[4]

L. Allen and T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), 197-232. doi: 10.1007/s002110100365.

[5]

B. Barker, J. Humpherys and K. Zumbrun, STABLAB: A MATLAB-based Numerical Library for Evans Function Computation Version 1. 0 preprint. Available from: http://impact.byu.edu/stablab/STABLAB_1.0_doc.pdf.

[6]

J. Billingham and D. J. Needham, A note on the properties of a family of traveling wave solutions arising in cubic autocatalysis, Dynam. Stability Systems, 6 (1991), 33-49. doi: 10.1080/02681119108806105.

[7]

T. J. BridgesG. Derks and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Phys. D, 172 (2002), 190-216. doi: 10.1016/S0167-2789(02)00655-3.

[8]

J. Carr, Application Of Centre Manifold Theory Springer-Verlag, New York-Berlin, 1981.

[9]

X. ChenY. Qi and Y. Zhang, Existence of traveling waves of auto-catalytic systems with decay, J. Differential Equations, 260 (2016), 7982-7999. doi: 10.1016/j.jde.2016.02.009.

[10]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations D. C. Heath and Co. , Boston, Mass, 1965.

[11]

S. Focant and Th. Galley, Existence and stability of propagating fronts for an autocatalytic reactio-diffuion systems, Physica D, 120 (1998), 346-368. doi: 10.1016/S0167-2789(98)00096-7.

[12]

S.-C. Fu and J.-C. Tsai, The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay, J. Differential Equations, 256 (2014), 3335-3364. doi: 10.1016/j.jde.2014.02.009.

[13]

P. GrayJ. F. Griffiths and S. K. Scott, Oscillations, Glow and Ignition in Carbon Monoxide Oxidation in an Open System Ⅰ. Experimental Studies of the Ignition Diagram and the Effects of Added Hydrogen, Proc. R. Soc. Lond. A, 397 (1985), 21-44. doi: 10.1098/rspa.1985.0002.

[14]

P. Gray, Instabilities and oscillations in chemical reactions in closed and open systems, Proc. R. Soc. Lond. A, 415 (1988), 1-34.

[15]

A. HannaA. Saul and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Am. Chem. Soc., 104 (1982), 3838-3844. doi: 10.1021/ja00378a011.

[16]

D. Henry, Geometric Theory Of Semilinear Parabolic Equations Springer-Verlag, Berlin-New York, 1981.

[17]

W. Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59. doi: 10.1016/j.jmaa.2005.04.084.

[18]

P. Howard and K. Zumbrun, The Evans functions and stbility criteria for degenerate viscous shcok waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855. doi: 10.3934/dcds.2004.10.837.

[19]

E. Jakab, D. Horvath, J. H. Merkin, S. K. Scott, P. L. Simon and A. Toth, Isothermal flame balls: Effect of autocatalyst decay Phys. Rev. E 66 (2002), 016207. doi: 10.1103/PhysRevE.66.016207.

[20]

Y. Li and Y. Wu, Stability of traveling waves with non-critical speeds for double degenerate Fisher-type equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 149-170. doi: 10.3934/dcdsb.2008.10.149.

[21]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), 1474-1521. doi: 10.1137/100814974.

[22]

Y. Li and Y. Wu, Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems, Math. Model. Nat. Phenom., 8 (2013), 104-132. doi: 10.1051/mmnp/20138308.

[23]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.I. Permanent form travelling waves, SIAM J. Appl. Math., 59 (1999), 870-899. doi: 10.1137/S0036139996312594.

[24]

P. M. McCabeJ. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay.Ⅱ. The initial boundary value problem, SIAM J. Appl. Math., 60 (2000), 1707-1748. doi: 10.1137/S0036139998344775.

[25]

J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system.Ⅳ. Quadratic autocatalysis with quadratic decay, Proc. R. Soc. Lond. A, 434 (1991), 531-554.

[26]

J. H. MerkinD. J. Needham and S. K. Scott, A simple model for sustained oscillations in isothermal branch-chain or autocatalytic reactions in a well stirred open system. Ⅰ. Stationary states and local stabilities, Proc. Roy. Soc. London Ser. A, 398 (1985), 81-100. doi: 10.1098/rspa.1985.0026.

[27]

J. H. MerkinD. J. Needham and S. K. Scott, The development of travelling waves in a simple isothermal chemical system.I. Quadratic autocatalysis with linear decay, Proc. Roy. Soc. London Ser. A, 424 (1989), 187-209. doi: 10.1098/rspa.1989.0075.

[28]

D. J. Needham, A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay, Z. Angew. Math. Phys., 42 (1991), 455-459. doi: 10.1007/BF00945715.

[29]

D. J. Needham and J. H. Merkin, The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Philos. Trans. Roy. Soc. London Ser. A, 337 (1991), 261-274. doi: 10.1098/rsta.1991.0122.

[30]

B. S. Ng and W. H. Reid, The compound matrix method for ordinary differential systems, J. Comput. Phys., 58 (1985), 209-228.

[31]

A. Pazy, Semigroups Of Linear Operators And Applications To Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[32]

B. P. Palka, An Introduction To Complex Function Theory Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0975-1.

[33]

R. L. PegoP. Smereka and M. I. Weinstein, Oscillatory instability of traveling waves for a KdV-Burgers equation, Phys. D, 67 (1993), 45-65. doi: 10.1016/0167-2789(93)90197-9.

[34]

R. L. Pego and M. I. Weinstein, Eigenvalues, and instability of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055.

[35]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes In C: The Art Of Scientific Computing. Second Edition Cambridge University Press, Cambridge, 1992.

[36]

L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs With MATLAB Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511615542.

[37]

Y. WuX. Xing and Q. Ye, Stability of travelling waves with algebraic decay for n-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 16 (2006), 47-66. doi: 10.3934/dcds.2006.16.47.

[38]

Q. Ye, Z. Y. Li, M. X. Wang and Y. Wu, Introduction To Reaction Diffusion Equation. Second Edition Science Press, Beijing, 2011.

Figure 1.  Wave profiles by using shooting method for $K=1.5$, $q=2$, $c=3>c^*$ (a) $d=1$, $u^*=1$, (b) $d=3$, $u^*=2.1853$
Figure 2.  Wave profiles for $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $c\!\!=\!\!3\!\!>\!\!c^*$, $K\!=\!1.5$, $q\!\!=\!\!2$, $sl\!\!=\!\!-500$, $sr\!\!=\!\!50$ and (a) $\xi_0\!\!=\!\!0$; (b) $\xi_0\!\!=\!\!-80$
Figure 3.  Wave profiles for $K\!\!=\!\!1.5$, $d\!\!=\!\!1$, $u^*\!\!=\!\!1$, $q\!=\!2$, $sr\!\!=\!\!50$, $\xi_0\!=\!0$ and (a) $sl\!\!=\!\!-500$ with different $c\!\!>\!\!c^*$; (b) $c\!\!=\!\!4$ with different starting point $sl$
Figure 4.  (a) The selected curve Γ for K=1:5, q=2, u=1, c=3 and d=1. (b) The numerical curves of E(Γ) for q = 1:5, K = 1:5 and d = 1 with different c > c
Figure 5.  The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $K\!\!>\!\!1$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
Figure 6.  The numerical curves of $E(\Gamma)$ for $q\!=\!2$ and $0\!\!<\!\!K\!\!\leq\!\!1$: (a) $K\!\!=\!\!0.5$, $c\!\!=\!\!3$ with different $d\!\in\!(0,d^*)$, (b) $d\!\!=\!\!1$, $K\!\!=\!\!0.5$ with different $c\!\!>\!\!c^*$
Figure 7.  The numerical curves of $E(\Gamma)$ for $1\!<q\!<2$: (a) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $q\!\!\in\!\!(1,2)$, (b) $q\!\!=\!\!1.5$, $K\!=\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$
Figure 8.  The numerical curves of $E(\Gamma)$ for $q=2.5$: (a) $K\!\!=\!\!1.5$, $c\!\!=\!\!4$ with different $d\!\in\!(0,d^*)$ (b) $d\!\!=\!\!1$, $K\!\!=\!\!1.5$ with different $c\!\!>\!\!c^*$
Table 1.  The values of $E(0)$ and $E(10^4)$ for $q=2$, $K>1$ with the parameters corresponding to the curves in Fig. 5, we just retain integers here, which is verified the estimates in Lemma 4.5
$K\!=\!1.5$, $c\!=\!3$$c\!=\!3$, $d\!=\!1$$K\!=\!1.5$, $d\!=\!1$
$d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$$K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$
$E(0)$109 77 23177 92 6878 66 136
$E(10^4)$49254 41166 2367241633 41262 4109341166 40480 40733
$K\!=\!1.5$, $c\!=\!3$$c\!=\!3$, $d\!=\!1$$K\!=\!1.5$, $d\!=\!1$
$d\!=\!0.7$ $d\!=\!1$ $d\!=\!3$$K\!=\!5$ $K\!=\!3$ $K\!=\!1.5$ $c\!=\!3$ $c\!=\!5$ $c\!=\!10$
$E(0)$109 77 23177 92 6878 66 136
$E(10^4)$49254 41166 2367241633 41262 4109341166 40480 40733
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