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On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras
Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China |
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.
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S. Ai,
Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.
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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. |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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