American Institute of Mathematical Sciences

Advanced
November  2007, 18(4): 613-626. doi: 10.3934/dcds.2007.18.613

An invariant set generated by the domain topology for parabolic semiflows with small diffusion

Nils Ackermann 1, , Thomas Bartsch 2, and  Petr Kaplický 3,

1. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, Mexico
2. 

Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany
3. 

Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic

Received  November 2006 Revised  February 2007 Published  May 2007

We consider the singularly perturbed semilinear parabolic problem $u_t-d^2\Delta u+u=f(u)$ with homogeneous Neumann boundary conditions on a smoothly bounded domain $\Omega\subseteq \mathbb{R}^N$. Here $f$ is superlinear at $0$, and $\pm\infty$ and has subcritical growth. For small $d>0$ we construct a compact connected invariant set $X_d$ in the boundary of the domain of attraction of the asymptotically stable equilibrium $0$. The main features of $X_d$ are that it consists of positive functions that are pairwise non-comparable, and its topology is at least as rich as the topology of $\partial\Omega$ in a certain sense. If the number of equilibria in $X_d$ is finite, then this implies the existence of connecting orbits within $X_d$ that are not a consequence of a well known result by Matano.
Keywords: Invariant Set, Singular Perturbation, Semilinear Parabolic Equation, Domain of Attraction., Connecting Orbit, Neumann Boundary Condition, Lusternik-Schnirelmann Category, Positive Equilibrium.
Mathematics Subject Classification: 35K55; 35K20, 35B25, 37B30, 55M3.
Citation: Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613
[1]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[2]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040
[3]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[4]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279
[5]

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

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[7]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297
[8]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280
[9]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[10]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
[11]

Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048
[12]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124
[13]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[14]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[15]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[16]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[17]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045
[18]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283
[19]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[20]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences