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November  2021, 29(5): 3017-3030. 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  November 2021 Early access  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, 2021, 29 (5) : 3017-3030. 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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