# American Institute of Mathematical Sciences

• Previous Article
Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice
• DCDS Home
• This Issue
• Next Article
Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media
July  2021, 41(7): 3141-3161. doi: 10.3934/dcds.2020401

## On some model problem for the propagation of interacting species in a special environment

 1 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland 2 School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China 3 Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

* Corresponding author: Mingmin Zhang

Received  April 2020 Revised  October 2020 Published  July 2021 Early access  December 2020

The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in domains of different dimensions together with some interaction of non classical type. We consider a truncated problem by imposing Dirichlet boundary conditions and an unbounded setting as well.

Citation: Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3141-3161. doi: 10.3934/dcds.2020401
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620-709.  doi: 10.1137/1018114. [2] H. Berestycki, J.-M. Roquejoffre and L. Rossi, The influence of a line with fast diffusion on Fisher-KPP propagation, J. Math. Biol., 66 (2013), 743-766.  doi: 10.1007/s00285-012-0604-z. [3] H. Berestycki, J.-M. Roquejoffre and L. Rossi, Fisher-KPP propagation in the presence of a line: Further effects, Nonlinearity, 26 (2013), 2623-2640.  doi: 10.1088/0951-7715/26/9/2623. [4] H. Berestycki, J.-M. Roquejoffre and L. Rossi, Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion, Nonlinear Analysis, 137 (2016), 171-189.  doi: 10.1016/j.na.2016.01.023. [5] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013. [6] M. Chipot, Elliptic Equations: An Introductory Course, Birkh$\ddot{ a }$user, Basel, Birkh$\ddot{ a }$user Advanced Texts, 2009. doi: 10.1007/978-3-7643-9982-5. [7] M. Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, London, 2016.  doi: 10.1142/p1064. [8] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 1, Masson, Paris, 1985. [9] L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Society, 2$^{nd}$ edition, 2010. doi: 10.1090/gsm/019. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [11] L. Rossi, A. Tellini and E. Valdinoci, The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary, SIAM J. Math. Anal., 49 (2017), 4595–4624. doi: 10.1137/17M1125388. [12] A. Tellini, Propagation speed in a strip bounded by a line with different diffusion, J. Differential Equations, 260 (2016), 5956-5986.  doi: 10.1016/j.jde.2015.12.028.

show all references

##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620-709.  doi: 10.1137/1018114. [2] H. Berestycki, J.-M. Roquejoffre and L. Rossi, The influence of a line with fast diffusion on Fisher-KPP propagation, J. Math. Biol., 66 (2013), 743-766.  doi: 10.1007/s00285-012-0604-z. [3] H. Berestycki, J.-M. Roquejoffre and L. Rossi, Fisher-KPP propagation in the presence of a line: Further effects, Nonlinearity, 26 (2013), 2623-2640.  doi: 10.1088/0951-7715/26/9/2623. [4] H. Berestycki, J.-M. Roquejoffre and L. Rossi, Travelling waves, spreading and extinction for Fisher-KPP propagation driven by a line with fast diffusion, Nonlinear Analysis, 137 (2016), 171-189.  doi: 10.1016/j.na.2016.01.023. [5] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013. [6] M. Chipot, Elliptic Equations: An Introductory Course, Birkh$\ddot{ a }$user, Basel, Birkh$\ddot{ a }$user Advanced Texts, 2009. doi: 10.1007/978-3-7643-9982-5. [7] M. Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, London, 2016.  doi: 10.1142/p1064. [8] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 1, Masson, Paris, 1985. [9] L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Society, 2$^{nd}$ edition, 2010. doi: 10.1090/gsm/019. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [11] L. Rossi, A. Tellini and E. Valdinoci, The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary, SIAM J. Math. Anal., 49 (2017), 4595–4624. doi: 10.1137/17M1125388. [12] A. Tellini, Propagation speed in a strip bounded by a line with different diffusion, J. Differential Equations, 260 (2016), 5956-5986.  doi: 10.1016/j.jde.2015.12.028.
The domain $\Omega_\ell$ for one-road problem
The domain $\Omega_\ell$ for two-road problem
The graph of the function $\rho(x_1)$
 [1] Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 [2] Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069 [3] N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079 [4] Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006 [5] Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025 [6] Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 [7] Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861 [8] Simone Göttlich, Camill Harter. A weakly coupled model of differential equations for thief tracking. Networks and Heterogeneous Media, 2016, 11 (3) : 447-469. doi: 10.3934/nhm.2016004 [9] Zhongwei Tang, Huafei Xie. Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (1) : 311-328. doi: 10.3934/cpaa.2020017 [10] Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i [11] Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control and Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353 [12] Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092 [13] Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050 [14] Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033 [15] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [16] Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184 [17] C. Bandle, Y. Kabeya, Hirokazu Ninomiya. Imperfect bifurcations in nonlinear elliptic equations on spherical caps. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1189-1208. doi: 10.3934/cpaa.2010.9.1189 [18] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 [19] Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205 [20] Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003

2021 Impact Factor: 1.588