American Institute of Mathematical Sciences

Advanced
March  2009, 2(1): 193-219. doi: 10.3934/dcdss.2009.2.193

Asymptotical dynamics of Selkov equations

Yuncheng You 1,

1. 

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620

Received  March 2008 Revised  July 2008 Published  January 2009

The existence of a global attractor for the solution semiflow of Selkov equations with Neumann boundary conditions on a bounded domain in space dimension $n\le 3$ is proved. This reaction-diffusion system features the oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The asymptotical compactness is shown by a new decomposition method. It is also proved that the Hausdorff dimension and fractal dimension of the global attractor are finite.
Keywords: global attractor, absorbing set, asymptotic compactness, asymptotical dynamics, fractal dimension., Selkov equation.
Mathematics Subject Classification: Primary: 37L30; Secondary: 35B40, 35B41, 35K55, 35K57, 35Q8.
Citation: Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193
[1]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[2]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
[3]

Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70.

[4]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[5]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[6]

Yuncheng You. Asymptotical dynamics of the modified Schnackenberg equations. Conference Publications, 2009, 2009 (Special) : 857-868. doi: 10.3934/proc.2009.2009.857
[7]

Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229
[8]

Emmanuel Hebey and Frederic Robert. Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electronic Research Announcements, 2004, 10: 135-141.

[9]

Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000
[10]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
[11]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281
[12]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[13]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[14]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[15]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495
[16]

Lorenzo Brasco, Marco Squassina, Yang Yang. Global compactness results for nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 391-424. doi: 10.3934/dcdss.2018022
[17]

Mohammad A. Safi, Abba B. Gumel. Global asymptotic dynamics of a model for quarantine and isolation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 209-231. doi: 10.3934/dcdsb.2010.14.209
[18]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[19]

Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[20]

Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences