NASCIMENTO, C. A.; http://lattes.cnpq.br/9723453546150758; NASCIMENTO, Cícero Alexandre do.
Abstract:
In this work we study the non local evolution problem
∂u(t, x)
∂t = −a(x)u(t, x) + K(f ◦ u)(t, x) + h(x, u(t, x)), t > 0, x ∈ Ω,
u(t, x) = 0, x ∈ Rn\ Ω,
u(0, x) = u0(x),
in a phase space isometric to Lp(Ω), where Ω is a smooth bounded domain in Rn. Here u = u(t, x) is a real value function, f : R −→ R is a continuously differentiable function, a ∈ W1,∞(Ω), h : R n × R −→ R is a continuously differentiable function with bounded derivative and K is an integral operator with a symmetric kernel. We prove the well posedness of problem, we prove the existence of global attractor and we exhibit a continuous Lyapunov functional for the flow generated by equation. Furthermore, using this Lyapunov functional we to prove that the flow is gradient and that there exists a non-trivial equilibrium solution. Finally, we study the upper semicontinuity of global attractors with respect to parameters a and h.