American Institute of Mathematical Sciences

Advanced
September  2008, 22(3): 465-508. doi: 10.3934/dcds.2008.22.465

Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain

Juncheng Wei 1, and  Jun Yang 2,

1. 

Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
2. 

Department of Mathematics, Shenzhen University, Shenzhen, China

Received  July 2007 Revised  March 2008 Published  August 2008

We consider the equation $\varepsilon^2\Delta$ ũ-ũ+ũ$^p =0$ in a bounded, smooth domain $\Omega$ in $\R^2$ under homogeneous Neumann boundary conditions. Let $\Gamma$ be a segment contained in $\Omega$, connecting orthogonally the boundary, non-degenerate and non-minimal with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.
Keywords: Gap condition, Clustered layers, singularly perturbed Neumann problem., Toda system.
Mathematics Subject Classification: Primary: 35B25, 35J60; Secondary: 35K99, 92C4.
Citation: Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465
[1]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[2]

Pingzheng Zhang, Jianhua Sun. Clustered layers for the Schrödinger-Maxwell system on a ball. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 657-688. doi: 10.3934/dcds.2006.16.657
[3]

Yang Wang. The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem. Communications on Pure and Applied Analysis, 2011, 10 (2) : 731-744. doi: 10.3934/cpaa.2011.10.731
[4]

Jaeyoung Byeon, Sang-hyuck Moon. Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1921-1965. doi: 10.3934/cpaa.2019089
[5]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[6]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709
[7]

Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[8]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
[9]

Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333
[10]

Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure and Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341
[11]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431
[12]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[13]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure and Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
[14]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[15]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[16]

Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823
[17]

Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1
[18]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041
[19]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499
[20]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy