Severino Horácio da Silva

Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

In this work we study the pullback dynamics of a class of nonlocal nonautonomous evolution equations for neural fields in a bounded smooth domain Ω in RN ∂tu(t, x) = −u(t, x) + ∫ RN J(x, y) f (t, u(t, y))dy, t ≥ τ, x ∈ Ω, u(τ, x) = uτ(x), x ∈ Ω, with u(t, x) = 0, t ≥ τ, x ∈ RN\Ω, where the integrable function J : RN ×RN → R is continuously differentiable, ∫ RN J(x, y)dy = ∫ RN J(x, y)dx = 1 and symmetric i.e., J(x, y) = J(y, x) for any x, y ∈ RN . Under suitable assumptions on the nonlinearity f : R2 → R, we prove existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to this problem.

Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

We show the normal hyperbolicity property for the equilibria of the evolution equation and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameter .