Global Attractor and Non-trivial Equilibrium Solution for Neural Fields with External Stimuli

In this paper we are interested in the study of the evolution problem \(\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u(t,x) &=- a(x)u(t,x) + K(f\circ u ) +h(x,u(t,x)) \ \ in \ \ \Omega ,\\ u(t,x) &= 0 \ \ in \ \ \mathbb {R}^N\backslash \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u(t,x)\hfill & =-a(x)u(t,x)+K(f\circ u)+h(x,u(t,x))in\mathrm{\Omega },\hfill \\ u(t,x)\hfill & =0in{\mathbb{R}}^N\backslash \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset \mathbb {R}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain with finite measure, \(f: \mathbb {R} \rightarrow \mathbb {R}\) $f:\mathbb{R}\to \mathbb{R}$ is a function of class \(C^{1}(\mathbb {R})\) $C^1(\mathbb{R})$ , \(a \in W^{1,\infty }(\Omega ),\) $a\in W^{1,\infty }(\mathrm{\Omega }),$ \(h:\mathbb {R}^{n} \times \mathbb {R} \longrightarrow \mathbb {R}\) $h:{\mathbb{R}}^n\times \mathbb{R}⟶\mathbb{R}$ is a continuously differentiable function with \(|\partial _{2}h|\le C_h\) $|{\partial }_{2}h|\le C_h$ , for some positive constant \(C_h\) $C_h$ and K is a operator given by \((Kv)(x)=\int _{\mathbb {R}^N}J(x,y)v(y)dy\) $(Kv)(x)={\int }_{{\mathbb{R}}^N}J(x,y)v(y)dy$ with \(J(x,y)=J(y,x)\) $J(x,y)=J(y,x)$ . We extend some results on global attractor and we show the existence of a Lyapunov functional. Furthermore, using the Lyapunov functional and the LaSalle’s Invariance Principle, we prove the existence of a non-trivial equilibrium solution. Finally, we prove that the flow continuously depends on the parameters a and h and show the upper semicontinuity of the attractors with respect to these parameters.