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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

Michel Chipot 1, and  Mingmin Zhang 2,3,,

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.

Keywords: Nonlinear elliptic systems, KPP equations, Coupled equations, Field-Road model, Truncated domain.
Mathematics Subject Classification: 35J47, 35J57.
Citation: Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013.  Google Scholar

[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.  Google Scholar

[7] M. Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, London, 2016.  doi: 10.1142/p1064.  Google Scholar
[8]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 1, Masson, Paris, 1985.  Google Scholar

[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.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013.  Google Scholar

[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.  Google Scholar

[7] M. Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, London, 2016.  doi: 10.1142/p1064.  Google Scholar
[8]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tome 1, Masson, Paris, 1985.  Google Scholar

[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.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The domain $ \Omega_\ell $ for one-road problem
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Figure 2.  The domain $ \Omega_\ell $ for two-road problem
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Figure 3.  The graph of the function $ \rho(x_1) $
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