A GRADIENT FLOW GENERATED BY A NONLOCAL MODEL OF A NEURAL FIELD IN AN UNBOUNDED DOMAIN

Author(s)

da Silva, Severino Horacio ^[1] ; Pereira, Antonio Luiz ^[2]

Affiliation

^[1] Univ Fed Campina Grande, UAMat, Ave Aprigio Veloso 882, Caixa Postal 10-044, BR-58109970 Campina Grande, PB - Brazil

^[2] Univ Sao Paulo, IME, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

• Document type: Journal article
• Source: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 51, n. 2, p. 583-598, JUN 2018.
• Web of Science Citations: 1

Abstract

In this paper we consider the nonlocal evolution equation Graphic We show that this equation defines a continuous flow in both the space C-b(R-N) of bounded continuous functions and the space C-rho(R-N) of continuous functions u such that u /s=b/ rho is bounded, where rho is a convenient ``weight function{''}. We show the existence of an absorbing ball for the flow in C-b(R-N) and the existence of a global compact attractor for the flow in C-rho(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(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. (AU)

• FAPESP's process: 16/02150-8 - Perturbation of domains and asymptotic behavior for boundary value problems
• Grantee: Antonio Luiz Pereira
• Support Opportunities: Regular Research Grants