Random dynamics of fractional stochastic retarded FitzHugh–Nagumo systems on unbounded domains

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 76 - 25
• Main Author: Gao, Dongjie
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 19.06.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Delay FitzHugh–Nagumo systems; Delay time; Dynamical systems; Fractional Laplacian operator; Mathematics; Mathematics and Statistics; Neurosciences; Partial differential equations; Pullback random attractor; Time dependence; Time-dependent property; Upper semicontinuity; Fractional Laplacian operator; 37L55; Upper semicontinuity; Pullback random attractor; Delay FitzHugh–Nagumo systems; 35B41; 35B40; Time-dependent property; 34K20
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03329-z

In this paper, we study the dynamics of fractional nonautonomous stochastic FitzHugh–Nagumo systems with variable delay defined on R n . We first consider the existence and uniqueness of pullback random attractors as well as the time-dependent properties of pullback random attractors. Then we study the upper semicontinuity of pullback random attractors as the delay time tends to zero. In order to achieve the goals of this article, we need to overcome two difficulties: (i) The noncompactness of Sobolev embedding on unbounded domains; (ii) The effect of the variable delay term. However, we can find the effective methods to resolve the two difficulties. More precisely, we use the method of backward uniform tail-estimates of solutions to resolve the first difficulty, and we introduce an inverse function to solve the second difficulty.