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March  2021, 14(3): 1047-1062. doi: 10.3934/dcdss.2020405

Fast reaction limit of reaction-diffusion systems

Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan

Received  January 2019 Revised  May 2020 Published  July 2020

Fund Project: This work was supported by JSPS KAKENHI Grant nos. 26287025, 15H03635 and 17K05368. Most of the work was performed during a visit of the author to Imperial College London thanks to JST CREST Grant No. JPMJCR14D3. The support of JST and the hospitality of Imperial College London are warmly acknowledged

Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.

Citation: Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
References:
[1]

D. AronsonM. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal, 6 (1982), 1001-1022.  doi: 10.1016/0362-546X(82)90072-4.  Google Scholar

[2]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135.  doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[3]

N. BouillardR. EymardM. HenryR. Herbin and D. Hilhorst, A fast precipitation and dissolution reaction for a reaction-diffusion system arising in a porous medium, Nonlinear Anal. Real World Appl., 10 (2009), 629-638.  doi: 10.1016/j.nonrwa.2007.10.019.  Google Scholar

[4]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math, 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.  Google Scholar

[5] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.   Google Scholar
[6]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[7]

J. EliašM. H. Kabir and M. Mimura, On the well-posedness of a dispersal model for farmers and hunter-gatherers in the neolithic transition, Math. Models Methods Appl. Sci, 28 (2018), 195-222.  doi: 10.1142/S0218202518500069.  Google Scholar

[8]

L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.   Google Scholar

[9]

R. EymardD. HilhorstH. Murakawa and M. Olech, Numerical approximation of a reaction-diffusion system with fast reversible reaction, Chinese Annals of Mathematics B, 31 (2010), 631-654.  doi: 10.1007/s11401-010-0604-5.  Google Scholar

[10]

R. Eymard, D. Hilhorst, R. van der Hout and L. A. Peletier, A reaction-diffusion system approximation of a one-phase Stefan problem, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 156–170.  Google Scholar

[11]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[12]

D. HilhorstJ. R. King and M. Röger, Mathematical analysis of a model describing the invasion of bacteria in burn wounds, Nonlinear Anal., 66 (2007), 1118-1140.  doi: 10.1016/j.na.2006.01.009.  Google Scholar

[13]

D. Hilhorst and H. Murakawa, Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium, Netw. Heterog. Media, 9 (2014), 669-682.  doi: 10.3934/nhm.2014.9.669.  Google Scholar

[14]

M. IidaH. MonobeH. Murakawa and H. Ninomiya, Vanishing, moving and immovable interfaces in fast reaction limits, J. Differential Equations, 263 (2017), 2715-2735.  doi: 10.1016/j.jde.2017.04.009.  Google Scholar

[15]

M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, Recent Advances on Elliptic and Parabolic Issues, eds. M. Chipot and H. Ninomiya, World Scientific, (2006), 145–164. Google Scholar

[16]

M. IidaH. Ninomiya and H. Yamamoto, A review on reaction-diffusion approximation, J. Elliptic Parabol. Equ., 4 (2018), 565-600.  doi: 10.1007/s41808-018-0029-y.  Google Scholar

[17] J. W. Jerome, Approximation of Nonlinear Evolution Systems, Academic Press, New York, 1983.   Google Scholar
[18]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[19]

P. Knabner, Mathematische Modelle für Transport und Sorption gelöster Stoffe in porösen Medien, Verlag Peter Lang, Frankfurt, 1991.  Google Scholar

[20]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[21]

A. MoussaB. Perthame and D. Salort, Backward parabolicity, cross-diffusion and turing instability, J. Nonlinear Sci., 29 (2019), 139-162.  doi: 10.1007/s00332-018-9480-z.  Google Scholar

[22]

H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.  doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[23]

H. Murakawa, A regularization of a reaction-diffusion system approximation to the two-phase Stefan problem, Nonlinear Anal., 69 (2008), 3512-3524.  doi: 10.1016/j.na.2007.09.038.  Google Scholar

[24]

H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete Contin. Dyn. Syst. S, 5 (2012), 147-158.  doi: 10.3934/dcdss.2012.5.147.  Google Scholar

[25]

H. Murakawa, A linear scheme to approximate nonlinear cross-diffusion systems, Math. Mod. Numer. Anal., 45 (2011), 1141-1161.  doi: 10.1051/m2an/2011010.  Google Scholar

[26]

H. Murakawa, An efficient linear scheme to approximate nonlinear diffusion problems, Jpn. J. Ind. Appl. Math., 35 (2018), 71-101.  doi: 10.1007/s13160-017-0279-3.  Google Scholar

[27]

H. Murakawa and H. Ninomiya, Fast reaction limit of a three-component reaction-diffusion system, J. Math. Anal. Appl., 379 (2011), 150-170.  doi: 10.1016/j.jmaa.2010.12.040.  Google Scholar

[28]

C. Verdi, Numerical aspects of parabolic free boundary and hysteresis problems, Lecture Notes in Mathematics, 1584 (1994), 213-284.  doi: 10.1007/BFb0073398.  Google Scholar

[29]

A. Visintin, Models of phase relaxation, Differential and Integral Equations, 14 (2001), 1469-1486.   Google Scholar

show all references

References:
[1]

D. AronsonM. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal, 6 (1982), 1001-1022.  doi: 10.1016/0362-546X(82)90072-4.  Google Scholar

[2]

D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135.  doi: 10.1016/S0022-247X(03)00457-8.  Google Scholar

[3]

N. BouillardR. EymardM. HenryR. Herbin and D. Hilhorst, A fast precipitation and dissolution reaction for a reaction-diffusion system arising in a porous medium, Nonlinear Anal. Real World Appl., 10 (2009), 629-638.  doi: 10.1016/j.nonrwa.2007.10.019.  Google Scholar

[4]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math, 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.  Google Scholar

[5] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.   Google Scholar
[6]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[7]

J. EliašM. H. Kabir and M. Mimura, On the well-posedness of a dispersal model for farmers and hunter-gatherers in the neolithic transition, Math. Models Methods Appl. Sci, 28 (2018), 195-222.  doi: 10.1142/S0218202518500069.  Google Scholar

[8]

L. C. Evans, A convergence theorem for a chemical diffusion-reaction system, Houston J. Math., 6 (1980), 259-267.   Google Scholar

[9]

R. EymardD. HilhorstH. Murakawa and M. Olech, Numerical approximation of a reaction-diffusion system with fast reversible reaction, Chinese Annals of Mathematics B, 31 (2010), 631-654.  doi: 10.1007/s11401-010-0604-5.  Google Scholar

[10]

R. Eymard, D. Hilhorst, R. van der Hout and L. A. Peletier, A reaction-diffusion system approximation of a one-phase Stefan problem, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 156–170.  Google Scholar

[11]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[12]

D. HilhorstJ. R. King and M. Röger, Mathematical analysis of a model describing the invasion of bacteria in burn wounds, Nonlinear Anal., 66 (2007), 1118-1140.  doi: 10.1016/j.na.2006.01.009.  Google Scholar

[13]

D. Hilhorst and H. Murakawa, Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium, Netw. Heterog. Media, 9 (2014), 669-682.  doi: 10.3934/nhm.2014.9.669.  Google Scholar

[14]

M. IidaH. MonobeH. Murakawa and H. Ninomiya, Vanishing, moving and immovable interfaces in fast reaction limits, J. Differential Equations, 263 (2017), 2715-2735.  doi: 10.1016/j.jde.2017.04.009.  Google Scholar

[15]

M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, Recent Advances on Elliptic and Parabolic Issues, eds. M. Chipot and H. Ninomiya, World Scientific, (2006), 145–164. Google Scholar

[16]

M. IidaH. Ninomiya and H. Yamamoto, A review on reaction-diffusion approximation, J. Elliptic Parabol. Equ., 4 (2018), 565-600.  doi: 10.1007/s41808-018-0029-y.  Google Scholar

[17] J. W. Jerome, Approximation of Nonlinear Evolution Systems, Academic Press, New York, 1983.   Google Scholar
[18]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[19]

P. Knabner, Mathematische Modelle für Transport und Sorption gelöster Stoffe in porösen Medien, Verlag Peter Lang, Frankfurt, 1991.  Google Scholar

[20]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[21]

A. MoussaB. Perthame and D. Salort, Backward parabolicity, cross-diffusion and turing instability, J. Nonlinear Sci., 29 (2019), 139-162.  doi: 10.1007/s00332-018-9480-z.  Google Scholar

[22]

H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332.  doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[23]

H. Murakawa, A regularization of a reaction-diffusion system approximation to the two-phase Stefan problem, Nonlinear Anal., 69 (2008), 3512-3524.  doi: 10.1016/j.na.2007.09.038.  Google Scholar

[24]

H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete Contin. Dyn. Syst. S, 5 (2012), 147-158.  doi: 10.3934/dcdss.2012.5.147.  Google Scholar

[25]

H. Murakawa, A linear scheme to approximate nonlinear cross-diffusion systems, Math. Mod. Numer. Anal., 45 (2011), 1141-1161.  doi: 10.1051/m2an/2011010.  Google Scholar

[26]

H. Murakawa, An efficient linear scheme to approximate nonlinear diffusion problems, Jpn. J. Ind. Appl. Math., 35 (2018), 71-101.  doi: 10.1007/s13160-017-0279-3.  Google Scholar

[27]

H. Murakawa and H. Ninomiya, Fast reaction limit of a three-component reaction-diffusion system, J. Math. Anal. Appl., 379 (2011), 150-170.  doi: 10.1016/j.jmaa.2010.12.040.  Google Scholar

[28]

C. Verdi, Numerical aspects of parabolic free boundary and hysteresis problems, Lecture Notes in Mathematics, 1584 (1994), 213-284.  doi: 10.1007/BFb0073398.  Google Scholar

[29]

A. Visintin, Models of phase relaxation, Differential and Integral Equations, 14 (2001), 1469-1486.   Google Scholar

Figure 1.  Level sets $ F(u,v) = 0 $, which are the graphs $ \{ (u,v)\ |\ v\in \alpha (u)\} $ (thick lines), and vector fields of (9) for some $ F $. (a) $ F(u,v) = F_2(u,-v) $ with $ F_2 $ of (8) (the limit of (1) is represented by the one-phase Stefan problem), (b) $ F $ of (11) with $ \beta $ of (10) (the limit of (1) is represented by the two-phase Stefan problem), (c) $ F $ of (11) with $ \beta $ of (12) (the limit of (1) is represented by the porous medium equation)
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