Jan W. Cholewa

Global Attractors in Abstract Parabolic Problems

Preface 1. Preliminary concepts 2. The abstract Cauchy problem 3. Semigroups of global solutions 4. Construction of the global attractor 5. Application of abstract results to parabolic equations 6. Examples of global attractors in parabolic problems 7. Backward uniqueness and regularity of solutions 8. Extensions 9. Appendix Bibliography Index.

LOCAL WELL POSEDNESS FOR STRONGLY DAMPED WAVE EQUATIONS WITH CRITICAL NONLINEARITIES

In this article the strongly damped wave equation is considered and a local well posedness result is obtained in the product space HQ(Q) X L 2 (il). The space of initial conditions is chosen according to the energy functional, whereas the approach used in this article is based on the theory of analytic semigroups as well as interpolation and extrapolation spaces. This functional analytic framework allows local existence results to be proved in the case of critically growing nonlinearities, which improves the existing results.

LINEAR PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp Lp-Lq-type estimates in uniform spaces for heat and Schrodinger semigroups and analyze the regularizing effect and the exponential type of these semigroups. We also deal with general second-order elliptic operators and study the generation of analytic semigoups in uniform spaces.

Global attractor for the Cahn-Hilliard system

The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space H = [.ff(Q)]. Following Hales ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of H. New difficulties specific to the system case make our problem interesting.

LOCAL WELL POSEDNESS, ASYMPTOTIC BEHAVIOR AND ASYMPTOTIC BOOTSTRAPPING FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS OF THE SECOND ORDER IN TIME

A class of semilinear evolution equations of the second order in time of the form u tt + Au + μAu t + Au tt = f(u) is considered, where -A is the Dirichlet Laplacian, Ω is a smooth bounded domain in R N and f ∈ C 1 (R,R). A local well posedness result is proved in the Banach spaces W 1,p 0 (Ω) x W 1,p 0 (Ω) when f satisfies appropriate critical growth conditions. In the Hilbert setting, if f satisfies an additional dissipativeness condition, the nonlinear semigroup of global solutions is shown to possess a gradient-like attractor. Existence and regularity of the global attractor are also investigated following the unified semigroup approach, bootstrapping and the interpolation-extrapolation techniques.

EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space , q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where and for certain . We construct a global attractor and show that is actually contained in an ordered interval [φm, φM], where is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of . A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.

Exponential global attractors for semigroups in metric spaces with applications to differential equations

In this article semigroups in a general metric space V , which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V . Applications of abstract results to ordinary and partial differential equations are given.

Reduction of Infinite Dimensional Systems to Finite Dimensions: Compact Convergence Approach

We consider parameter dependent semilinear evolution problems for which, at the limit value of the parameter, the problem is finite dimensional. We introduce an abstract functional analytic framework that applies to many problems in the existing literature for which the study of asymptotic dynamics can be reduced to finite dimensions via the invariant manifolds technique. Some practical models are considered to show wide applicability of the theory.

Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation

We consider a family of bounded dissipative asymptotically compact semigroups depending on a parameter, and study the continuity properties of the corresponding family of its global attractors. We exploit the idea of the uniform exponential attraction property to discuss the continuity properties of the family of attractors and estimate the rate of convergence of the approximating attractors to the limit one. Showing a range of applications of an abstract framework, we focus much of our attention on a perturbed damped wave equation. In this latter case our results involve nonlinearities with critical exponents, for which the continuity of the family of attractors is concluded, including the rate of convergence and the regularity of the limit attractor. This complements the results in the literature.

Parabolic equations with critical nonlinearities

As well known the problem of global continuation of solutions to semilinear parabolic equations is completely solved when the nonlinear term is subordinated to an