- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 541426, 18 pages
Dimension Estimate for the Global Attractor of an Evolution Equation
Departamento de Matemática, Pontificia Universidad Javeriana, Carrera 7 No. 43-82, Bogotá, Colombia
Received 29 September 2011; Accepted 13 December 2011
Academic Editor: Juan J. Nieto
Copyright © 2012 Renato Colucci and Gerardo R. Chacón. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer, New York, NY, USA, 2nd edition, 1997.
- J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001.
- L. Yang and M.-H. Yang, “Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition,” Nonlinear Analysis, vol. 11, no. 5, pp. 3946–3954, 2010.
- T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact non-autonomous dynamical systems,” Nonlinear Analysis, vol. 64, no. 3, pp. 484–498, 2006.
- A. Tarasińska, “Pullback attractor for heat convection problem in a micropolar fluid,” Nonlinear Analysis, vol. 11, no. 3, pp. 1458–1471, 2010.
- M. Yang and P. E. Kloeden, “Random attractors for stochastic semi-linear degenerate parabolic equations,” Nonlinear Analysis, vol. 12, no. 5, pp. 2811–2821, 2011.
- Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains,” Nonlinear Analysis, vol. 12, no. 6, pp. 3468–3482, 2011.
- T. Caraballo, J. A. Langa, V. S. Melnik, and J. Valero, “Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,” Set-Valued Analysis, vol. 11, no. 2, pp. 153–201, 2003.
- O. V. Kapustyan, P. O. Kasyanov, and J. Valero, “Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 535–547, 2011.
- N. Saito, “Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,” IMA Journal of Numerical Analysis, vol. 27, no. 2, pp. 332–365, 2007.
- M. Efendiev, E. Nakaguchi, and W. L. Wendland, “Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme,” Journal of Mathematical Analysis and Applications, vol. 358, no. 1, pp. 136–147, 2009.
- A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37, John Wiley & Sons, 1995.
- M. Bulíček and D. Pražák, “A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1025–1028, 2009.
- G. Bellettini, G. Fusco, and N. Guglielmi, “A concept of solution and numerical experiments for forward-backward diffusion equations,” Discrete and Continuous Dynamical Systems A, vol. 16, no. 4, pp. 783–842, 2006.
- G. R. Chacón and R. Colucci, “Asymptotic behavior of a fourth order evolution equation,” submitted paper.
- P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, vol. 70 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1st edition, 1988.
- J. C. Robinson, Dimensions, Embeddings, and Attractors, vol. 186 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2011.
- M. Slemrod, “Dynamics of measure valued solutions to a backward-forward heat equation,” Journal of Dynamics and Differential Equations, vol. 3, no. 1, pp. 1–28, 1991.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, NY, USA, 2010.
- R. Mañé, “On the dimension of the compact invariant sets of certain nonlinear maps,” in Dynamical Systems and Turbulence, vol. 898 of Lecture Notes in Mathematics, pp. 230–242, Springer, Berlin, Germany, 1981.
- C. Foias and R. Temam, “Determination of the solutions of the Navier-Stokes equations by a set of nodal values,” Mathematics of Computation, vol. 43, no. 167, pp. 117–133, 1984.