doi: 10.3934/era.2020116

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

School of Science, Chang'an University, Xi'an, Shaanxi 710064, China

* Corresponding author: Shao-Xia Qiao

Received  November 2019 Revised  July 2020 Published  November 2020

Fund Project: The program was partially supported by NSF of China (11671180)

This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.

Citation: Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, doi: 10.3934/era.2020116
References:
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P. W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13–52.  Google Scholar

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J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

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F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

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W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

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W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

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W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

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Y.-J. SunW.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonnlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[23]

Y.-J. SunW.-T. Li and Z.-C. Wang, Entire solutions for nonlocal dispersal equations with bistable nonlineartiy, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

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[25]

J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[26]

S.-L. Wu and S. Ruan, Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case, J. Differential Equations, 258 (2015), 2435-2470.  doi: 10.1016/j.jde.2014.12.013.  Google Scholar

[27]

H. Yagisita, Existence of traveling wave solutions for a nonlocal bistable equation: An abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.  doi: 10.2977/prims/1260476649.  Google Scholar

[28]

L. ZhangW.-T. Li and Z.-C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804.  doi: 10.1007/s11425-016-9003-7.  Google Scholar

[29]

L. ZhangW.-T. LiZ.-C. Wang and Y.-J. Sun, Entire solutions in nonlocal bistable equations: Asymmetric case, Acta Math. Sin. (Engl. Ser.), 35 (2019), 1771-1794.  doi: 10.1007/s10114-019-8294-8.  Google Scholar

show all references

References:
[1]

P. W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13–52.  Google Scholar

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.  Google Scholar

[3]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.  Google Scholar

[4]

H. BerestyckiF. Hamel and H. Matano, Bistable traveling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275.  Google Scholar

[5]

H. Berestycki and N. Rodríguez, A non-local bistable reaction-diffusion equaiton with a gap, Discrete Contin. Dyn. Syst., 37 (2017), 685-723.  doi: 10.3934/dcds.2017029.  Google Scholar

[6]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[7]

X. Chen, Existence, uniqueness and asymptotic stablility of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[8]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

[9]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect., 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[10]

S. Eberle, A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities, J. Differential Equations, 265 (2018), 804-829.  doi: 10.1016/j.jde.2018.03.007.  Google Scholar

[11]

S. Eberle, A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section, Nonlinear Anal., 172 (2018), 99-114.  doi: 10.1016/j.na.2018.03.004.  Google Scholar

[12]

P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin (2003), 153–191.  Google Scholar

[13]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[14]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[15]

W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[16]

W.-T. LiJ.-B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[17]

T. S. Lim and A. Zlatoš, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602.  Google Scholar

[18]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[19]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.  Google Scholar

[20]

W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

[21]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[22]

Y.-J. SunW.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonnlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[23]

Y.-J. SunW.-T. Li and Z.-C. Wang, Entire solutions for nonlocal dispersal equations with bistable nonlineartiy, J. Differential Equations, 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[24]

M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayes, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[25]

J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[26]

S.-L. Wu and S. Ruan, Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case, J. Differential Equations, 258 (2015), 2435-2470.  doi: 10.1016/j.jde.2014.12.013.  Google Scholar

[27]

H. Yagisita, Existence of traveling wave solutions for a nonlocal bistable equation: An abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.  doi: 10.2977/prims/1260476649.  Google Scholar

[28]

L. ZhangW.-T. Li and Z.-C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804.  doi: 10.1007/s11425-016-9003-7.  Google Scholar

[29]

L. ZhangW.-T. LiZ.-C. Wang and Y.-J. Sun, Entire solutions in nonlocal bistable equations: Asymmetric case, Acta Math. Sin. (Engl. Ser.), 35 (2019), 1771-1794.  doi: 10.1007/s10114-019-8294-8.  Google Scholar

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