Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria

Gheorghe Craciun; Jiaxin Jin; Casian Pantea; Adrian Tudorascu

doi:10.3934/dcdsb.2020164

Article Contents

2021, Volume 26, Issue 3: 1305-1335. Doi: 10.3934/dcdsb.2020164

This issue Previous Article Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum Next Article Bistability of sequestration networks

Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria

1.
Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison
2.
Department of Mathematics, University of Wisconsin-Madison
3.
Department of Mathematics, West Virginia University

Received: May 31, 2019

Revised: January 31, 2020

Early access: May 2020

Published: March 2021

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.

Keywords:

Mathematics Subject Classification: 35B40, 35K57, 35Q92, 80A30, 80A32, 90E20.

Citation:

Full Text(HTML)

Figure 1. Construction of a rectangular invariant region for the reversible reaction $ m_1A+n_1B\rightleftharpoons m_2A+n_2B $ for the cases a. $ \bar m = m_1-m_2, \ \bar n = n_2-n_1 $ nonzero and of the same sign; b. $ \bar m, \bar n $ nonzero and of different signs

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	D. F. Anderson, Global asymptotic stability for a class of nonlinear chemical equations, SIAM J. Appl. Math., 68 (2008), 1464-1476. doi: 10.1137/070698282.
[2]	D. F. Anderson and A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858. doi: 10.1137/090764098.
[3]	D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508. doi: 10.1137/11082631X.
[4]	A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100. doi: 10.1081/PDE-100002246.
[5]	W. X. Chen, C. M. Li and E. S. Wright, On a nonlinear parabolic system-modeling chemical reactions in rivers, Communications On Pure And Applied Analysis, 4 (2005), 889-899. doi: 10.3934/cpaa.2005.4.889.
[6]	M. Choulli, L. Kayser and E. M. Ouhabaz, Observations on Gaussian upper bounds for Neumann heat kernels, Bulletin of the Australian Mathematical Society, 92 (2015), 429-439. doi: 10.1017/S0004972715000611.
[7]	G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565. doi: 10.1016/j.jsc.2008.08.006.
[8]	G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329. doi: 10.1137/100812355.
[9]	G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2016), arXiv: 1501.02860.
[10]	L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176. doi: 10.1016/j.jmaa.2005.07.003.
[11]	L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431. doi: 10.4171/RMI/541.
[12]	L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104.
[13]	L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Adv. Nonlinear Stud., 7 (2007), 491-511. doi: 10.1515/ans-2007-0309.
[14]	L. Desvillettes, K. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion system arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709. doi: 10.1137/16M1073935.
[15]	M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972/73), 187-194. doi: 10.1007/BF00255665.
[16]	K. Fellner, W. Prager and B. Q. Tang, The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks, Kinet. Relat. Models, 10 (2017), 1055–1087, arXiv: 1504.08221. doi: 10.3934/krm.2017042.
[17]	K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., 69 (2018), Paper No. 54, 30 pp. doi: 10.1007/s00033-018-0948-3.
[18]	J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (2015), 553-587. doi: 10.1007/s00205-015-0866-x.
[19]	W. E. Fitzgibbon, J. Morgan and R. Sanders, Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems, Nonlin. Anal., 19 (1992), 885-899. doi: 10.1016/0362-546X(92)90057-L.
[20]	M. Gopalkrishnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797. doi: 10.1137/130928170.
[21]	M. Gopalkrishnan, E. Miller and A. Shiu, A projection argument for differential inclusions, with applications to persistence of mass-action kinetics, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 025, 25 pp. doi: 10.3842/SIGMA.2013.025.
[22]	F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116. doi: 10.1007/BF00251225.
[23]	F. Horn, The dynamics of open reaction systems, Mathematical Aspects of Chemical and Biochemical Problems and Quantum Chemistry, SIAM-AMS Proceedings, Amer. Math. Soc., Providence, R.I., 8 (1974), 125-137.
[24]	O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs, AMS, 23 (1995).
[25]	A. Mielke, J. Haskovec and P. A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928. doi: 10.1007/s10884-014-9394-x.
[26]	M. Minchevaand and D. Siegel, Stability of mass action reaction-diffusion systems, Nonlinear Anal., 56 (2004), 1105-1131. doi: 10.1016/j.na.2003.10.025.
[27]	F. Mohamed, C. Pantea and A. Tudorascu, Chemical reaction-diffusion networks: Convergence of the method of lines, J. Math. Chem., 56 (2018), 30-68. doi: 10.1007/s10910-017-0779-z.
[28]	C. Pantea, On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673. doi: 10.1137/110840509.
[29]	M. Pierre, T. Suzuki and H. Umakoshi, Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria, J. Appl. Anal. Comp., 8 (2018), 836-858. doi: 10.11948/2018.836.
[30]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[31]	F. Rothe, Global Solutions of Reaction-Diffusion System, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.
[32]	D. Siegel and D. MacLean, Global stability of complex balanced mechanisms, J. Math. Chem., 27 (2000), 89-110. doi: 10.1023/A:1019183206064.
[33]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.
[34]	E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE Trans. Automat. Control, 46 (2001), 1028-1047. doi: 10.1109/9.935056.
[35]	M. E. Taylor, Partial Differential Equation Ⅲ. Nonlinear Equations, Springer Series Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
[36]	P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L^p$-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51. doi: 10.1090/S1079-6762-02-00104-X.