American Institute of Mathematical Sciences

Advanced
  • 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

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  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The domain $ \Omega_\ell $ for two-road problem
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The graph of the function $ \rho(x_1) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
[2]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[3]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069
[4]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[5]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2653-2676. doi: 10.3934/dcds.2020379
[6]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[7]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021043
[8]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[9]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
[10]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[11]

Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078
[12]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
[13]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009
[14]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[15]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[16]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[17]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022
[18]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
[19]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[20]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (42)
  • HTML views (150)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences