Boundary blow-up solutions with interior layers and spikes in a bistable problem

Previous Article
Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
DCDS Home
This Issue
Next Article
Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation

2007, 19(2): 271-298. doi: 10.3934/dcds.2007.19.271

Boundary blow-up solutions with interior layers and spikes in a bistable problem

Yihong Du ^1, , Zongming Guo ^2, and Feng Zhou ^3,

1.	School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351
2.	Department of Mathematics, Henan Normal University, Xinxiang, 453007, China
3.	Department of Mathematics, East China Normal University, Shanghai 200062

Received May 2005 Revised November 2005 Published July 2007

Abstract
Related Papers

We show that for small $\epsilon>0$, the boundary blow-up problem

$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

Keywords: spike layer, reduction method., bistable, boundary blow-up, Boundary and interior layer.

Mathematics Subject Classification: 35J20, 35J6.

Citation: Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271

[1]	Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333
[2]	Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[3]	Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929
[4]	Gung-Min Gie, Chang-Yeol Jung, Roger Temam. Recent progresses in boundary layer theory. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2521-2583. doi: 10.3934/dcds.2016.36.2521
[5]	X. Liang, Roderick S. C. Wong. On a Nested Boundary-Layer Problem. Communications on Pure & Applied Analysis, 2009, 8 (1) : 419-433. doi: 10.3934/cpaa.2009.8.419
[6]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[7]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[8]	Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[9]	Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1
[10]	Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[11]	Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[12]	D. Sanchez. Boundary layer on a high-conductivity domain. Communications on Pure & Applied Analysis, 2002, 1 (4) : 547-564. doi: 10.3934/cpaa.2002.1.547
[13]	Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
[14]	Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333
[15]	Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[16]	Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537
[17]	Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[18]	Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555
[19]	Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[20]	Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113

2016 Impact Factor: 1.099

American Institute of Mathematical Sciences