TY - JOUR
AU - da Silva, Severino Horacio
AU - Pereira, AntÃ´nio Luiz
PY - 2018/05/23
Y2 - 2024/04/16
TI - A gradient flow generated by a nonlocal model of a neutral field in an unbounded domain
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 51
IS - 2
SE -
DO -
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2018.004
SP - 583 - 598
AB - In this paper we consider the nonlocal evolution equation\[\frac{\partial u(x,t)}{\partial t} + u(x,t)=\int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)dy+ h(x).\]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 theequilibrium point set. We also illustrate our result with aconcrete example.
ER -