Global Attractors in Abstract Parabolic Problems

Abstract

The study of dissipative equations is an area that has attracted substantial attention over many years. Much progress has been achieved using a combination of both finite dimensional and infinite dimensional techniques, and in this book the authors exploit these same ideas to investigate the asymptotic behaviour of dynamical systems corresponding to parabolic equations. In particular the theory of global attractors is presented in detail. Extensive auxiliary material and rich references make this self-contained book suitable as an introduction for graduate students, and experts from other areas, who wish to enter this field.

Dissipative parabolic equations in locally uniform spaces

“…[9,18,19,45], a functional analytic setting and a consequent analytic semigroup approach in locally uniform spaces constitute in the present paper a main tool both to obtain local well posedness result and to derive sufficiently smooth estimates of the solutions. Such an approach is used in the monographs [26,2,31,14] and the present paper remains in the same vein.…”

“…For example in [9,14] the assumption on the nonlinear term reads f (x, u) = −λ 0 u + f 0 (u) + g(x), with λ 0 > 0, f 0 (u)u ≤ 0, (…”

“…In the last fifteen years a number of problems in unbounded domains originating in applications have been intensively studied [1,6,9,12,13,14,15,16,18,19,20,23,32,33,34,45]. The references contain merely a sample list of the related publications.…”

On Global Attractors for a Nonlinear Parabolic Equation of m-Laplacian Type in RN

“…[3,5,6,16]) and the parabolic equations of m-Laplacian type is one of the most typical nonlinear evolution equations, and so our argument and result here seem to be useful to consider other related problems.…”

“…The existence and some properties of global attractors of the nonlinear parabolic equations of m-Laplacian type in bounded domains have been studied by several authors (see Cholewa and Dlotko [5], Takeuchi and Yokota [15], Matsuura and Otani [9] and Nakao and Aris [11] etc., in this case the term lu is not necessary). But, there seem to be little results for the problem in R N .…”