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

In this paper, we study some questions related to attractors for two types of reaction-diffusion equations : an equation with a polynomial growth nonlinearity and systems admitting a positively invariant region. For these problems, we prove the existence of a maximal attractor which describes the long-time behaviour of the solutions and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our results are applied to several classical equations.