American Institute of Mathematical Sciences

Advanced
February  2005, 12(2): 193-212. doi: 10.3934/dcds.2005.12.193

Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations

Jong-Shenq Guo 1, and  Yoshihisa Morita 2,

1. 

Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 117, Taiwan
2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan

Received  July 2003 Revised  October 2003 Published  December 2004

We study entire solutions of a scalar reaction-diffusion equation of 1-space dimension. Here the entire solutions are meant by solutions defined for all $(x,t)\in\mathbb R^2$. Assuming that the equation has traveling front solutions and using the comparison argument, we prove the existence of entire solutions which behave as two fronts coming from the both sides of $x$-axis. A key idea for the proof of the main results is to characterize the asymptotic behavior of the solutions as $t\to-\infty$ in terms of appropriate subsolutions and supersolutions. This argument can apply not only to a general bistable reaction-diffusion equation but also to the Fisher-KPP equation. We also extend our argument to the Fisher-KPP equation with discrete diffusion.
Keywords: Fisher-KPP equation, discrete diffusion., traveling front, bistable nonlinearity, Reaction-diffusion equation, entire solution.
Mathematics Subject Classification: 35K57, 35B05, 35B40, 37L6.
Citation: Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193
[1]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[2]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[3]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868
[4]

Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801
[5]

Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516
[6]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[7]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[8]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
[9]

Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1
[10]

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
[11]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21
[12]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011
[13]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126
[14]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
[15]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[16]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087
[17]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[18]

Pengchao Lai, Qi Li. Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021186
[19]

Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11
[20]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (151)
  • HTML views (0)
  • Cited by (96)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences