# American Institute of Mathematical Sciences

February  2021, 14(2): 455-464. 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, 2021, 14 (2) : 455-464. 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] 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, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164 [2] Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 [3] Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 [4] Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049 [5] Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019 [6] Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283 [7] Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329 [8] Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334 [9] Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 [10] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [11] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [12] Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126 [13] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [14] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [15] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [16] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [17] Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004 [18] Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 [19] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376 [20] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

2019 Impact Factor: 1.233

Article outline