American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations
  • ERA Home
  • This Issue
  • Next Article
    On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras
doi: 10.3934/era.2021024

Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect

Meng-Xue Chang , Bang-Sheng Han , and  Xiao-Ming Fan

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

* Corresponding author: Bang-Sheng Han

Received  December 2020 Revised  February 2021 Published  March 2021

Fund Project: The second author is supported by NSF grant11801470 and the Fundamental Research Funds for the Central Universities (2682018CX64)

This paper is devoted to studying the Cauchy problem corresponding to the nonlocal bistable reaction diffusion equation. It is the first attempt to use the method of comparison principle to study the well-posedness for the nonlocal bistable reaction-diffusion equation. We show that the problem has a unique solution for any non-negative bounded initial value by using Gronwall's inequality. Moreover, the boundedness of the solution is obtained by means of the auxiliary problem. Finally, in the case that the initial data with compactly supported, we analyze the asymptotic behavior of the solution.

Keywords: Reaction-diffusion, nonlocal, existence, asymptotic.
Mathematics Subject Classification: Primary: 35A01, 35K55, 35K57, 92D25; Secondary: 35K57.
Citation: Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, doi: 10.3934/era.2021024
References:
[1]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[2]

M. Alfaro, N. Apreutesei, F. Davidson and V. Volpert, Preface to the issue nonlocal reaction-diffusion equations, Math. Model. Nat. Phenom., 10 (2015), 1-5.  doi: 10.1051/mmnp/201510601.  Google Scholar

[3]

M. Alfaro and J. Coville, Rapid travelling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[4]

M. Alfaro, J. Coville and G. Raoul, Bistable travelling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.  doi: 10.3934/dcds.2014.34.1775.  Google Scholar

[5]

N. Apreutesei, A. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

[6]

X. Bao and W. -T. Li, Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats, Nonlinear Anal. Real World Appl., 51 (2020), 102975, 26 pp. doi: 10.1016/j. nonrwa. 2019.102975.  Google Scholar

[7]

N. Bessonov, N. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

[8]

J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.  Google Scholar

[9]

I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101.  doi: 10.1051/mmnp/20105506.  Google Scholar

[10]

K. Deng and Y. Wu, Global stability for a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006.  Google Scholar

[11]

G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006.  Google Scholar

[12]

S. Génieys, V. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, J. Comptes Rendus Biologies, 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.  Google Scholar

[13]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[14]

B. -S. Han, M. -X. Chang and Y. Yang, Spatial dynamics of a nonlocal bistable reaction diffusion equation, Electron. J. Differential Equations, (2020), Paper No. 84, 23 pp.  Google Scholar

[15]

B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959-1983.  doi: 10.3934/dcdsb.2020011.  Google Scholar

[16]

B.-S. Han and Y.-H. Yang, On a predator-prey reaction-diffusion model with nonlocal effects, Commun. Nonlinear Sci. Numer. Simul., 46 (2017), 49-61.  doi: 10.1016/j.cnsns.2016.10.018.  Google Scholar

[17]

B.-S. Han and Y. Yang, An integro-PDE model with variable motility, Nonlinear Anal. Real World Appl., 45 (2019), 186-199.  doi: 10.1016/j.nonrwa.2018.07.004.  Google Scholar

[18]

B. -S. Han, Y. Yang, W. -J. Bo and H. Tang, Global dynamics of a Lotka-Volterra competition diffusion system with nonlocal effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050066, 19 pp. doi: 10.1142/S0218127420500662.  Google Scholar

[19]

J. Li, E. Latos and L. Chen, Wavefronts for a nonlinear nonlocal bistable reaction-diffusion equation in population dynamics, J. Differential Equations, 263 (2017), 6427-6455.  doi: 10.1016/j.jde.2017.07.019.  Google Scholar

[20]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[21]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.  doi: 10.1137/050638011.  Google Scholar

[22]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.  Google Scholar

[23]

Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of travelling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

show all references

References:
[1]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[2]

M. Alfaro, N. Apreutesei, F. Davidson and V. Volpert, Preface to the issue nonlocal reaction-diffusion equations, Math. Model. Nat. Phenom., 10 (2015), 1-5.  doi: 10.1051/mmnp/201510601.  Google Scholar

[3]

M. Alfaro and J. Coville, Rapid travelling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[4]

M. Alfaro, J. Coville and G. Raoul, Bistable travelling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.  doi: 10.3934/dcds.2014.34.1775.  Google Scholar

[5]

N. Apreutesei, A. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

[6]

X. Bao and W. -T. Li, Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats, Nonlinear Anal. Real World Appl., 51 (2020), 102975, 26 pp. doi: 10.1016/j. nonrwa. 2019.102975.  Google Scholar

[7]

N. Bessonov, N. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

[8]

J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.  Google Scholar

[9]

I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101.  doi: 10.1051/mmnp/20105506.  Google Scholar

[10]

K. Deng and Y. Wu, Global stability for a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006.  Google Scholar

[11]

G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006.  Google Scholar

[12]

S. Génieys, V. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, J. Comptes Rendus Biologies, 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.  Google Scholar

[13]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[14]

B. -S. Han, M. -X. Chang and Y. Yang, Spatial dynamics of a nonlocal bistable reaction diffusion equation, Electron. J. Differential Equations, (2020), Paper No. 84, 23 pp.  Google Scholar

[15]

B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959-1983.  doi: 10.3934/dcdsb.2020011.  Google Scholar

[16]

B.-S. Han and Y.-H. Yang, On a predator-prey reaction-diffusion model with nonlocal effects, Commun. Nonlinear Sci. Numer. Simul., 46 (2017), 49-61.  doi: 10.1016/j.cnsns.2016.10.018.  Google Scholar

[17]

B.-S. Han and Y. Yang, An integro-PDE model with variable motility, Nonlinear Anal. Real World Appl., 45 (2019), 186-199.  doi: 10.1016/j.nonrwa.2018.07.004.  Google Scholar

[18]

B. -S. Han, Y. Yang, W. -J. Bo and H. Tang, Global dynamics of a Lotka-Volterra competition diffusion system with nonlocal effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050066, 19 pp. doi: 10.1142/S0218127420500662.  Google Scholar

[19]

J. Li, E. Latos and L. Chen, Wavefronts for a nonlinear nonlocal bistable reaction-diffusion equation in population dynamics, J. Differential Equations, 263 (2017), 6427-6455.  doi: 10.1016/j.jde.2017.07.019.  Google Scholar

[20]

G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[21]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.  doi: 10.1137/050638011.  Google Scholar

[22]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.  Google Scholar

[23]

Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of travelling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[1]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471
[2]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[3]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[4]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157
[5]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
[6]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
[7]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[8]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[9]

Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
[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]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[12]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[13]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114
[14]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
[15]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427
[16]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
[17]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415
[18]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[19]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817
[20]

Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (37)
  • HTML views (58)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences