Attractors for reaction–diffusion equations in \[$\mathbb{R}^{N}$ with continuous nonlinearity

Article type: Research Article

Authors: Morillas, Francisco | Valero, José

Affiliations: Universidad Politécnica de Valencia, ETSI Geodésica, Cartográfica y Topográfica, Camino de Vera, s/n, 46022 Valencia, Spain | Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain E-mail: jvalero@umh.es

Abstract: In this paper we prove the existence of a compact global attractor for a reaction–diffusion equation on \[$\mathbb{R}^{N}$. We do not assume that the nonlinear term is differentiable (just continuous) and, also, we do not guarantee the uniqueness of solutions of the Cauchy problem. Besides, the growth and dissipative conditions are different from the ones used in previous papers on the topic. An application is given to the Fitz–Hugh–Nagumo system, which models the transmission of signals across axons.

Keywords: reaction–diffusion equations, set-valued dynamical system, global attractor, unbounded domain

Journal: Asymptotic Analysis, vol. 44, no. 1-2, pp. 111-130, 2005