American Institute of Mathematical Sciences

Advanced
September  2010, 26(3): 901-921. doi: 10.3934/dcds.2010.26.901

The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment

Shin-Ichiro Ei 1, and  Hiroshi Matsuzawa 2,

1. 

Faculty of Mathematics, Kyushu University, Hakozaki 6-10-1, Higashiku, Fukuoka-city, Fukuoka 812-8560, Japan
2. 

Numazu National College of Technology, Ooka 3600, Numazu-city, Shizuoka 410-8501

Received  February 2009 Revised  November 2009 Published  December 2009

In this paper we study the dynamics of a single transition layer of a solution to a spatially inhomogeneous bistable reaction diffusion equation in one space dimension. The spatial inhomogeneity is given by a function $a(x)$. In particular, we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$. In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics. In order to analyze such a dynamics, we construct an attractive local invariant manifold giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold. We also give applications of our results to two well known nonlinearities of bistable type.
Keywords: transition layer; bistable; reaction diffusion equation..
Mathematics Subject Classification: Primary: 35B25; Secondary: 35K5.
Citation: Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901
[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5627-5640. doi: 10.3934/dcdsb.2020370
[8]

Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124
[9]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
[10]

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
[11]

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
[12]

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
[13]

François Hamel, Jean-Michel Roquejoffre. Heteroclinic connections for multidimensional bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 101-123. doi: 10.3934/dcdss.2011.4.101
[14]

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
[15]

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
[16]

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
[17]

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
[18]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[19]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[20]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy