A non-autonomous model for neuronal dynamical: A relationship between neuronal activity and external stimulus. (arXiv:1712.09751v1 [math.DS])

In this work we consider a class of nonlocal non-autonomous evolution problems, which arise in neuronal activity, \[ \begin{cases} \partial_t u(t,x) =- a(t)u(t,x) + b(t) \displaystyle\int_{\mathbb{R}^N} J(x,y)f(t,u(t,y))dy -h +S(t,x) ,\ t\geq\tau \in \mathbb{R},\ x \in \Omega, u(\tau,x)=u_\tau(x),\ x\in\Omega u(t,x)= 0,\ t\geq\tau,\ x \in\mathbb{R}^N\backslash\Omega. \end{cases} \] Under suitable assumptions on the nonlinearity $f: \mathbb{R} \times \mathbb{R} \to\mathbb{R}$ and constraints on the functions $J: \mathbb{R}^N \times \mathbb{R}^{N}\to \mathbb{R}$;\, $S: \mathbb{R} \times \mathbb{R}^{N}\to\mathbb{R}$ and $a,b:\mathbb{R} \to\mathbb{R}$, we study the assimptotic behavior of the evolution process, generated by this problem, in an appropriated Banach space.% and we present a brief discussion on the model with a biological interpretation. We prove results on existence, uniqueness and smoothness of the solutions and the existence of pullback attracts for the evolution process as 查看全文>>