-
Previous Article
Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains
- DCDS Home
- This Issue
-
Next Article
Stochastic matrix-valued cocycles and non-homogeneous Markov chains
Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation
| 1. | Department of Mathematics I, Wüllnerstr. 5-7, RWTH Aachen, 52056 Aachen, Germany |
$ u_t=a(x) (\Delta u+\lambda_1 u) \qquad $ (*)
with zero Dirichlet data in a smoothly bounded domain $\Omega \subset \R^n$, $n\ge 1$. Here $a$ is positive in $\Omega$ and Hölder continuous in $\bar\Omega$, and $\lambda_1>0$ denotes the principal eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet data. It is shown that if $\int_\Omega \frac{(\dist(x,\partial\Omega))^2}{a(x)}dx=\infty$ then there exist initial data in $W^{1,\infty}(\Omega)$ such that the solution of (*) is bounded but not convergent as $t\to\infty$: It has a totally ordered $\omega$-limit set which is not a singleton. Under the above condition, the occurrence of even unbounded ordered $\omega$-limit sets is demonstrated. Conversely, if $\frac{(\dist(x,\partial\Omega))^2}{a(x)}$ is integrable then any solution emanating from initial data in $W^{1,\infty}(\Omega)$ converges to some stationary solution of (*) as time approaches infinity.
| [1] |
Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487 |
| [2] |
Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 |
| [3] |
Qianqian Han, Bo Deng, Xiao-Song Yang. The existence of $ \omega $-limit set for a modified Nosé-Hoover oscillator. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022043 |
| [4] |
Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 |
| [5] |
Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819 |
| [6] |
Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065 |
| [7] |
Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787 |
| [8] |
José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781 |
| [9] |
Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979 |
| [10] |
Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413 |
| [11] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
| [12] |
Francisco Balibrea, J.L. García Guirao, J.I. Muñoz Casado. A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 983-994. doi: 10.3934/dcds.2002.8.983 |
| [13] |
Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223 |
| [14] |
Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096 |
| [15] |
José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010 |
| [16] |
Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141 |
| [17] |
Jaroslav Smítal, Marta Štefánková. Omega-chaos almost everywhere. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1323-1327. doi: 10.3934/dcds.2003.9.1323 |
| [18] |
Lan Wen. On the preperiodic set. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237 |
| [19] |
François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262 |
| [20] |
V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]