American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 516-525. doi: 10.3934/proc.2009.2009.516

On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments

Hiroshi Matsuzawa 1,

1. 

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

Received  July 2008 Revised  June 2009 Published  September 2009

In this paper we consider a boundary value problem for the following semilinear elliptic equation :$-\varepsilon^{2}\Delta u=h(|x|)^2(u-a(|x|))(1-u^2)$ in $B_1(0)$ with homogeneous Neumann boundary condition. The function $a$ is a $C^1$ function satisfying $|a(r)|< 1$ for $r\in [0,1]$ and $a^'(0)=0$. The function $h$ is a positive $C^1$ function satisfying $h^'(0)=0$. The nonlinear function in the equation is a typical example of the so-called {\it bistable} nonlinearity. Functions $a$ and $h$ in the nonlinearity represent spatial inhomogeneity. In particular, we consider the case where $a(r)=0$ on some interval $I\subset (0,1)$. When $\varepsilon>0$ is very small, there exist stationary solutions with sharp transition layers. We investigate asymptotic locations of transition layers of a global minimizer corresponding to an energy functional as $\varepsilon\to 0$. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function $h$.
Keywords: bistable equation, transition layer, unbalanced, Allen-Cahn equation.
Mathematics Subject Classification: Primary:35K57, 35J50,; Secondary:35J25.
Citation: 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
[1]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[2]

Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402
[3]

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
[4]

Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021030
[5]

Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021032
[6]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073
[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[9]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029
[10]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015
[11]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[14]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[15]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[16]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384
[17]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065
[18]

José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025
[19]

Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037
[20]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

 Impact Factor: 

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences