Details

Autor(en) / Beteiligte

• Wang, Bingqi
• Zhou, Xiangyu

Titel

Single peak solutions for an elliptic system of FitzHugh–Nagumo type

Ist Teil von

• Journal of fixed point theory and applications, 2024-06, Vol.26 (2), Article 13

Ort / Verlag

Cham: Springer International Publishing

Erscheinungsjahr

2024

Quelle

SpringerLink

Beschreibungen/Notizen

• We study the Dirichlet problem for an elliptic system derived from FitzHugh–Nagummo model as follows: - ε 2 Δ u = f ( u ) - v , in Ω , - Δ v + γ v = δ ε u , in Ω , u = v = 0 , on ∂ Ω , where Ω represents a bounded smooth domain in R 2 and ε , γ are positive constants. The parameter δ ε > 0 is a constant dependent on ε , and the nonlinear term f ( u ) is defined as u ( u - a ) ( 1 - u ) . Here, a is a function in C 2 ( Ω ) ∩ C 1 ( Ω ¯ ) with its range confined to ( 0 , 1 2 ) . Our research focuses on this spatially inhomogeneous scenario whereas the scenario that a is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov–Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on a , which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment.