Propriedade gradiente para equação de campos neurais com força externa variável.

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.