Abstract
The close-to-equilibrium regularity of solutions to a class of reaction–diffusion systems is investigated. The considered systems typically arise from chemical reaction networks and satisfy a complex balanced condition. Under some restrictions on spatial dimensions (\(d\le 4\)) and order of nonlinearities (\(\mu = 1 + 4/d \)), we show that if the initial data are close to a complex balanced equilibrium in \(L^2\)-norm, then classical solutions are shown global and converging exponentially to equilibrium in \(L^{\infty }\)-norm. Possible extensions to higher dimensions and order of nonlinearities are also discussed. The results of this paper improve the recent work (Cáceres and Cañizo in Nonlinear Anal TMA 159:62–84, 2017).
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References
H. Amann. “Global existence for semilinear parabolic systems.” J. Reine Angew. Math. 360 (1985): 47–83.
J.A. Cañizo, L. Desvillettes and K. Fellner. “Improved duality estimates and applications to reaction-diffusion equations.” Comm. Partial Differential Equations 39.6 (2014): 1185–1204.
M.C. Caputo and A. Vasseur. “Global regularity of solutions to systems of reaction diffusion with sub-quadratic growth in any dimension.” Comm. Partial Differential Equations 34.10 (2009): 1228–1250.
M.J. Cáceres and J.A. Cañizo. “Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance.” Nonlinear Anal. 159 (2017): 62–84.
L. Desvillettes, K. Fellner and B.Q. Tang, “Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks”, SIAM J. Math. Anal. 49.4 (2017): 2666–2709.
L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle. Global existence for quadratic systems of reaction-diffusion. Adv. Nonlinear Stud. 7(3), (2007) 491–511.
L. Evans. “Partial differential equations”, American Mathematical Society, 2nd Edition (2010).
M. Feinberg, “Lectures on Chemical Reaction Networks”, University of Wisconsin-Madison, 1979, https://crnt.osu.edu/LecturesOnReactionNetworks.
M. Feinberg, “Chemical reaction network structure and the stability of complex isothermal reactors. I. The deficiency zero and deficiency one theorems”, Chem. Eng. Sci., 42 (1987), pp. 2229–2268.
K. Fellner and E.H. Laamri. “Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems” J. Evol. Equ. (2016) 16: 681.
K. Fellner and B.Q. Tang, “Convergence to equilibrium for renormalised solutions to nonlinear chemical reaction-diffusion systems”, arXiv:1708.01427.
K. Fellner, E. Latos and T. Suzuki. “Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions.” Discrete Contin. Dyn. Syst. Ser. B 2016, 21, 3441–3462.
J. Fischer. “Global existence of renormalized solutions to entropy-dissipating reaction diffusion systems.” Arch. Ration. Mech. Anal. 218.1 (2015): 553–587.
A. Glitzky and R. Hunlich. “Global estimates and asymptotics for electro reaction diffusion systems in heterostructures.” Appl. Anal. 66.3-4 (1997): 205–226.
T. Goudon and A. Vasseur. “Regularity analysis for systems of reaction-diffusion equations.” Ann. Sci. École Norm. Sup. Vol. 43. No. 1. Elsevier, 2010.
O.A. Ladyzhenskaya and N.N. Ural’tseva. “Linear and quasilinear equations of parabolic type.” Transaction of Mathematical Monographs, American Mathematical Society, Vol. 23 (1988).
F. Horn and R. Jackson. “General mass action kinetics.” Arch. Ration. Mech. Anal. 47.2 (1972): 81–116.
A. Mielke. “Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics”. to appear in Springer Proceedings in Mathematics & Statistics, WIAS preprint 2326.
M. Pierre. “Weak solutions and supersolutions in \(L^1\) for reaction-diffusion systems.” Nonlinear Evolution Equations and Related Topics. Birkhuser Basel, 2003. 153–168.
M. Pierre, D. Schmitt. “Blowup in reaction-diffusion systems with dissipation of mass.” SIAM Review 42.1 (2000): 93–106.
M. Pierre, “Global existence in reaction-diffusion systems with control of mass: a survey”,Milan J. Math. 78.2 (2010): 417–455.
M. Pierre, T. Suzuki and R. Zou. “Asymptotic behavior of solutions to chemical reaction-diffusion systems.” J. Math. Anal. Appl. 450.1 (2017): 152–168.
F. Rothe. “Global solutions of reaction-diffusion systems.” Lecture Notes in Mathematics, Vol. 1072, Springer 1984.
D. Siegel and M.D. Johnston, “Linearization of complex balanced chemical reaction systems” Preprint 2008.
B.Q. Tang, “Global classical solutions to reaction-diffusion systems in one and two dimensions”. to appear in Commun. Math. Sci.
Acknowledgements
Open access funding provided by University of Graz. We would like to thank the referee for helpful suggestions which improve the presentation of the paper.
This work is partially supported by International Training Program IGDK 1754 and NAWI Graz.
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Tang, B.Q. Close-to-equilibrium regularity for reaction–diffusion systems. J. Evol. Equ. 18, 845–869 (2018). https://doi.org/10.1007/s00028-017-0422-2
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DOI: https://doi.org/10.1007/s00028-017-0422-2