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Applicable Analysis
An International Journal
Volume 25, 1987 - Issue 1-2
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Original Articles

Attractors for reaction-diffusion equations: existence and estimate of their dimension

Martine Marion Laboratoire d'Analyse Numérique, Université Paris-Sud, Bâtiment 425, Orsay Cédex, 91405, France
Pages 101-147 | Received 24 Mar 1987, Published online: 02 May 2007

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Ehmet Ablet, Xiaoling Chen, Lixin Cheng, Quanqing Fang & Zheming Zheng. (2021) On measure of noncompactness and application to global attractors of operator semigroups. Quaestiones Mathematicae 44:1, pages 73-88.
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Xuntian Yan, Xianming Liu & Meihua Yang. (2017) Random attractors of stochastic partial differential equations: A smooth approximation approach. Stochastic Analysis and Applications 35:6, pages 1007-1029.
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A. Debussche. (1997) On the finite dimensionality of random attractors . Stochastic Analysis and Applications 15:4, pages 473-491.
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M. W. Smiley. (1993) Global attractors and approximate inertial. Applicable Analysis 50:3-4, pages 217-241.
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Joel D. Avrin & Mokhtar Kirane. (1993) Temperature growth and temperature bounds in special cases of combustion models. Applicable Analysis 50:1-2, pages 131-144.
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D. Brochet, D. Hilhorst & Xinfu chen. (1993) Finite dimensional exponential attractor for the phase field model. Applicable Analysis 49:3-4, pages 197-212.
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Oscar Manley & Martine Marion. (1993) Attractor Dimension for a Simple Premixed Flame Propagation Model. Combustion Science and Technology 88:1-2, pages 15-32.
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Jie Shen. (1989) Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations. Numerical Functional Analysis and Optimization 10:11-12, pages 1213-1234.
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W. E. Fitzgibbon, M. Langlais & J. J. Morgan. (1999) Eventually uniform bounds for a class of quasipositive reaction diffusion systems. Japan Journal of Industrial and Applied Mathematics 16:2, pages 225-241.
Crossref
A. Debussche. (1998) Hausdorff dimension of a random invariant set. Journal de Mathématiques Pures et Appliquées 77:10, pages 967-988.
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Joel Avrin. (1997) Incompressible reacting flows. Transactions of the American Mathematical Society 349:10, pages 3875-3892.
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Francisco Balibrea & Jos? Valero. (1997) Estimates of dimension of attractors of reaction-diffusion equations in the non-differentiable case. Comptes Rendus de l'Acad?mie des Sciences - Series I - Mathematics 325:7, pages 759-764.
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Alexandre N Carvalho, Sergio M Oliva, Antônio L Pereira & Anı́bal Rodriguez-Bernal. (1997) Attractors for Parabolic Problems with Nonlinear Boundary Conditions. Journal of Mathematical Analysis and Applications 207:2, pages 409-461.
Crossref
Russell A. Smith. (2011) Poincaré–Bendixson theory for certain reaction–diffusion boundary-value problems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124:1, pages 33-69.
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Roger Temam. 1994. Trends and Perspectives in Applied Mathematics. Trends and Perspectives in Applied Mathematics 293 336 .
Joel D. Avrin. (1993) Asymptotic Behavior of One-Step Combustion Models with Multiple Reactants on Bounded Domains. SIAM Journal on Mathematical Analysis 24:2, pages 290-298.
Crossref
Olivier Goubet. (1993) Nonlinear galerkin methods using hierarchical almost-orthogonal finite elements bases. Nonlinear Analysis: Theory, Methods & Applications 20:3, pages 223-247.
Crossref
A Debussche & M Marion. (1992) On the construction of families of approximate inertial manifolds. Journal of Differential Equations 100:1, pages 173-201.
Crossref
Aníbal Rodríguez-Bernal. (1992) Initial value problem and asymptotic low dimensional behavior in the Kuramoto-Velarde equation. Nonlinear Analysis: Theory, Methods & Applications 19:7, pages 643-685.
Crossref
Tepper L. Gill & W. W. Zachary. (1992) Dimensionality of Invariant Sets for Nonautonomous Processes. SIAM Journal on Mathematical Analysis 23:5, pages 1204-1229.
Crossref
Joseph W.-H. So & Jianhong Wu. (2009) Topological dimensions of global attractors for semilinear PDE's with delays. Bulletin of the Australian Mathematical Society 43:3, pages 407-422.
Crossref
Joel D. Avrin. (1991) Qualitative Theory of the Cauchy Problem for a One-Step Reaction Model on Bounded Domains. SIAM Journal on Mathematical Analysis 22:2, pages 379-391.
Crossref
Keith Promislow. (1991) Time analyticity and gevrey regularity for solutions of a class of dissipative partial differential equations. Nonlinear Analysis: Theory, Methods & Applications 16:11, pages 959-980.
Crossref
A. Eden, B. Michaux & J. M. Rakotoson. (1991) Doubly nonlinear parabolic-type equations as dynamical systems. Journal of Dynamics and Differential Equations 3:1, pages 87-131.
Crossref
A. Eden, B. Michaux & J.M. Rakotoson. (1990) Some results on doubly nonlinear parabolic equations as dynamical systems. Applied Mathematics Letters 3:1, pages 5-8.
Crossref
Frederic Abergel. (1990) Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. Journal of Differential Equations 83:1, pages 85-108.
Crossref
Martine Marion. (1989) Finite-Dimensional Attractors Associated with Partly Dissipative Reaction-Diffusion Systems. SIAM Journal on Mathematical Analysis 20:4, pages 816-844.
Crossref
Martine Marion. (1989) Approximate inertial manifolds for reaction-diffusion equations in high space dimension. Journal of Dynamics and Differential Equations 1:3, pages 245-267.
Crossref
Martine Marion & Roger Temam. 1988. Mathematical Modeling in Combustion and Related Topics. Mathematical Modeling in Combustion and Related Topics 155 172 .

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