• 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

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.

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. AlfaroN. ApreuteseiF. 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. AlfaroJ. 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. ApreuteseiA. 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. BessonovN. 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. CovilleJ. 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. GénieysV. 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. HanZ.-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. LiE. 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. NadinB. 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. WangW.-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. AlfaroN. ApreuteseiF. 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. AlfaroJ. 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. ApreuteseiA. 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. BessonovN. 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. CovilleJ. 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. GénieysV. 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. HanZ.-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. LiE. 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. NadinB. 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. WangW.-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]

Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249

[2]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[5]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081

[6]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[7]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021085

[8]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217

[9]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001

[10]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400

[11]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[12]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

[13]

Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021064

[14]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[15]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067

[16]

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

[17]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403

[18]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129

[19]

Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$ ^+ $-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021090

[20]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385

 Impact Factor: 0.263

Article outline

[Back to Top]