A gradient flow generated by a nonlocal model of a neural field in an unbounded domain

2017/04/17 by Severino H. da Silva, da Silva, Severino Horácio, Antônio Luíz Pereira +1
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation

paper · pdf · doi:10.48550/arxiv.1704.04938

Abstract

In this paper we consider the non local evolution equation (∂ u(x,t))/(∂ t) + u(x,t)= ∫_ℝ^NJ(x-y)f(u(y,t))ρ(y)dy+ h(x). % h ≥ 0. 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_ρ(ℝ^N) of continuous functions u such that u ⋅ ρ is bounded, where ρ 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_ρ(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_ρ(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.

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