Dissipative property for a class of non local evolution equations

In this work we consider the non local evolution problem $$\{\begin{array}{l}{\mathrm{\partial }}_{t}u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h),x\in \mathrm{\Omega },t\in [0,\mathrm{\infty }[;\\ u(x,t)=0,x\in {\mathbb{R}}^N\setminus \mathrm{\Omega },t\in [0,\mathrm{\infty }[;\\ u(x,0)=u_0(x),x\in {\mathbb{R}}^N,\end{array}$$ where $\mathrm{\Omega }$ is a smooth bounded domain in $$\mathbb{R}$^N, ~g,f: $\mathbb{R}$\to$\mathbb{R}$$satisfyingcertaingrowingconditionand$K$ is an integral operator with symmetric kernel, $Kv(x)={\int }_{{\mathbb{R}}^N}J(x,y)v(y)dy\mathrm{.}$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov’s functional, concluding that the flow generated by this equation has a gradient property.