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- W2950010200 abstract "In this paper we consider the non local evolution equation $$ frac{partial u(x,t)}{partial t} + u(x,t)= int_{mathbb{R}^{N}}J(x-y)f(u(y,t))rho(y)dy+ h(x). %,,, h geq 0. $$ We show that this equation defines a continuous flow in both the space $C_{b}(mathbb{R}^{N})$ of bounded continuous functions and the space $C_{rho}(mathbb{R}^{N})$ of continuous functions $u$ such that $u cdot rho$ is bounded, where $rho $ is a convenient weight function'. We show the existence of an absorbing ball for the flow in $C_{b}(mathbb{R}^{N})$ and the existence of a global compact attractor for the flow in $C_{rho}(mathbb{R}^{N})$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is well defined in the whole phase space and continuous in the $C_{rho}(mathbb{R}^{N})$ topology, allowing the characterization of the attractor as the unstable set of the equilibrium point set. We also illustrate our result with a concrete example." @default.
- W2950010200 created "2019-06-27" @default.
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- W2950010200 date "2017-04-17" @default.
- W2950010200 modified "2023-09-23" @default.
- W2950010200 title "A gradient flow generated by a nonlocal model of a neural field in an unbounded domain" @default.
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