Figure 1.
The domain $ \Omega_\ell $ for one-road problem
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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 |
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 - A, doi: 10.3934/dcds.2020401
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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.
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. |
| [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.
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. |
| [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. |
Figure 2.
The domain $ \Omega_\ell $ for two-road problem
Figure 3.
The graph of the function $ \rho(x_1) $
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