Symmetric patterns of a delayed reaction-diffusion equation with nonlinear boundary condition

Xiaowei Qu; Shangjiang Guo

doi:10.3934/dcdsb.2026022

Article Contents

2026, Volume 36: 1-24. Doi: 10.3934/dcdsb.2026022

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Symmetric patterns of a delayed reaction-diffusion equation with nonlinear boundary condition

Xiaowei Qu and
Shangjiang Guo^,

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

^*Corresponding author: Shangjiang Guo
^*Corresponding author: Shangjiang Guo

Received: August 14, 2025

Early access: December 31, 2025

Published online: December 31, 2025

The second author is supported by the National Natural Science Foundation of P.R. China (Grant No. 12471173), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant No. CUGST2).

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Abstract

This paper is concerned with the dynamical behaviors of radially symmetric steady-state solutions for a delayed reaction-diffusion equation with nonlinear boundary condition on a unit disk. By means of Lyapunov-Schmidt reduction, we derive the existence of radially symmetric steady-state solutions, and also obtain equivariant Hopf bifurcation near the steady-state solutions. Furthermore, symmetry-breaking happens when periodic solutions bifurcate from the radially symmetric steady-state solutions. For illustration, we apply our theoretical results to a Nicholson's blowflies system with nonlinear boundary condition.

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Mathematics Subject Classification: Primary: 34K20, 35B36; Secondary: 35K61.

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