Nonlinear Analysis 75 (2012) 5702–5722 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na An estimate on the fractal dimension of attractors of gradient-like dynamical systems M.C. Bortolan a , T. Caraballo b,∗ , A.N. Carvalho a , J.A. Langa b a Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil b Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 - Sevilla, Spain article info Article history: Received 12 March 2012 Accepted 21 May 2012 Communicated by Enzo Mitidieri Keywords: Fractal dimension Morse decomposition Gradient-like semigroups Evolution process abstract This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, A ∗ ) is an attractor–repeller pair for the attractor A of a semigroup {T (t ) : t ≥ 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A ∗ , the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction Over the last forty years, the study of qualitative properties of semigroups in Banach spaces has received very much attention (see, for instance, [1–5]). In particular, the study of global attractors has created a deep area of research and greatly improved the understanding of qualitative properties of solutions for these infinite dimensional dynamical systems. A particular aspect that has called the attention of many researchers, and for which a very nice theory has been developed, is the fractal dimension of attractors. Starting with the pioneering works [6,7], the theory has grown considerably and new strategies to find bounds for the fractal dimension have been proposed (see for example [5,8,4,2] and references therein). Before we proceed, let us briefly recall the definitions of topological, Hausdorff and fractal dimension. If K is a topological space, we say that K has a finite topological dimension if there exists a natural number d such that, for every open covering U of K , there is another covering U ′ of K refining U with the property that each point of K belongs to at most d + 1 sets in U ′ . In this case, the topological dimension dim T (K ) of K is the minimum d with this property. With this notion, a subset of R n with non-empty interior has topological dimension n and, if K is a compact metric space with topological dimension dim T (K )< ∞, then it is homeomorphic to a subset of R n with n = 2 dim T (K ) + 1 (see [9,10]). ∗ Corresponding author. Tel.: +34 954557998; fax: +34 954552898. E-mail addresses: matheusb@icmc.usp.br (M.C. Bortolan), caraball@us.es (T. Caraballo), andcarva@icmc.usp.br (A.N. Carvalho), langa@us.es (J.A. Langa). 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.05.018