Semilinear cooperative elliptic systems on Rn

We study here the following semilinear cooperative elliptic system defined on IRn , n > 2 : (1 – a) −∆u = aρ(x)u + bρ(x)v + f(x, u, v) x ∈ IRn , (1 – b) −∆v = cρ(x)u + dρ(x)v + g(x, u, v) x ∈ IRn , (1 – c) u −→ 0 , v −→ 0 as |x| −→ +∞. Here a, b, c, d are constants such that b, c > 0 ; ρ, f and g are given functions; ρ is nonnegative and tends to 0 at ∞. We first establish necessary and sufficient conditions on the coefficients for having a Maximum Principle for the linear System. Then we show that these conditions ensure existence of solutions for the linear System and for the semilinear System when f and g satisfy some ”sublinear” condition. Under some additional assumption we also derive uniqueness of the solutions. Finally we show that our results can be extended to N × N systems, N > 2

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techniques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.Comment: 39 page

The 'mechanism' of human cognitive variation

The theory of psychosis and autism as diametrical disorders offers a tractable and testable view of normal and abnormal human cognitive variation as a function of opposing traits grouped by their selection for maternal and paternal reproductive fitness. The theory could be usefully rooted and developed with reference to the lower-level perceptual and attentional phenomena from which social cognitive modules are developmentally refined

Monotone Maps: a review

This paper is dedicated to Jim Cushing on the occasion of his 62th birthday The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics