In this work we prove the existence of a compact global attractor for the flow of the equationin L 2 (S 1 ). We also give uniform estimates on the size of the attractor and show that the family of attractors {A J } is upper semicontinuous at J 0 .

No abstract

Abstract. In this work we prove the existence of a compact global attractor for the flow of the equationin L 2 (S 1 ). We also give uniform estimates on the size of the attractor and show that the flow is gradient.

In this work, we consider the invariant manifolds for the family of equationsẋ = Ax + f (ε, x), where A the is generator of a strongly continuous semigroup of linear operators in a Banach space X and f (ε, •) : X → X is continuous. The existence of stable (unstable) and center-stable (center-unstable) manifolds for a large class of these equations has been proved in [2]. We prove here that, if A admits a exponential trichotomy and f satisfies some suitable regularity hypotheses, then those manifolds are continuous with respect to the parameter ε.

In this work we consider the non local evolution problemwhere Ω is a smooth bounded domain in R N ; g, f : R → R satisfying certain growing condition and K is an integral operator with symmetric kernel, Kv(x) = R N J(x, y)v(y)dy. We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.