doi: 10.3934/dcdss.2020355

Stationary reaction-diffusion systems in $ L^1 $ revisited

1. 

Université de Lorraine, B.P. 239, 54506 Vandœuvre-lès-Nancy, France

2. 

Ecole Normale Supérieure de Rennes and IRMAR, Campus de Ker Lann, 35170-Bruz, France

* Corresponding author: El Haj Laamri

Received  December 2019 Published  May 2020

We prove existence of $ L^1 $-weak solutions to the reaction-diffusion system obtained as a stationary version of the system arising for the evolution of concentrations in a reversible chemical reaction, coupled with space diffusion. This extends a previous result by the same authors where restrictive assumptions on the number of chemical species are removed.

Citation: El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020355
References:
[1]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (4) (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar

[2]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (1) (2015), 553-587. doi: 10.1007/s00205-015-0866-x.  Google Scholar

[3]

E. H. Laamri and M. Pierre, Stationary reaction-diffusion systems in $L^1$, M3AS, 28 (11) (2018), 2161-2190. doi: 10.1142/S0218202518400110.  Google Scholar

[4]

R. H. Martin and M. Pierre, Influence of mixed boundary conditions in some reaction-diffusion systems, Proc. Roy. Soc. Edinburgh, Sect. A, 127 (1997), 1053-1066. doi: 10.1017/S0308210500026883.  Google Scholar

[5]

M. Pierre, Global existence in reaction-diffusion systems with control of mass : a survey, Milan. J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

show all references

References:
[1]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (4) (1973), 565-590. doi: 10.2969/jmsj/02540565.  Google Scholar

[2]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Ration. Mech. Anal., 218 (1) (2015), 553-587. doi: 10.1007/s00205-015-0866-x.  Google Scholar

[3]

E. H. Laamri and M. Pierre, Stationary reaction-diffusion systems in $L^1$, M3AS, 28 (11) (2018), 2161-2190. doi: 10.1142/S0218202518400110.  Google Scholar

[4]

R. H. Martin and M. Pierre, Influence of mixed boundary conditions in some reaction-diffusion systems, Proc. Roy. Soc. Edinburgh, Sect. A, 127 (1997), 1053-1066. doi: 10.1017/S0308210500026883.  Google Scholar

[5]

M. Pierre, Global existence in reaction-diffusion systems with control of mass : a survey, Milan. J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

[1]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042

[2]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020164

[3]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[4]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[5]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[6]

Congming Li, Eric S. Wright. Modeling chemical reactions in rivers: A three component reaction. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 377-384. doi: 10.3934/dcds.2001.7.373

[7]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189

[8]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[9]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[10]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815

[11]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891

[12]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733

[13]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[14]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[15]

Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843

[16]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106

[17]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921

[18]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[19]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[20]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

2019 Impact Factor: 1.233

Article outline

[Back to Top]