Existence and Stability Theory of a Neurologically Inspired Parabolic PDE Model with a Nonlinear Time-Delayed Boundary Condition

Authors

Gangadhara Boregowda
Michael R. Lindstrom, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

9-20-2025

Abstract

In this paper, we establish the existence of a positive, bounded solution for a class of parabolic partial differential equations with nonlinear boundary conditions, where the boundary conditions depend on the solution on the boundary at a time 𝜏≥0 in the past. These equations model the production dynamics of a protein species by a single cell, where a feedback mechanism downregulates the protein’s production. Furthermore, we analyze the stability of a nontrivial steady-state solution and provide sufficient conditions on the nonlinearity parameter, boundary flux, and time-delay that ensure the occurrence of a Hopf bifurcation.

Recommended Citation

Boregowda, Gangadhara, and Michael R. Lindstrom. "Existence and Stability Theory of a Neurologically Inspired Parabolic PDE Model with a Nonlinear Time-Delayed Boundary Condition." International Journal of Bifurcation and Chaos (2025): 2550170. https://doi.org/10.1142/S0218127425501706

Publication Title

International Journal of Bifurcation and Chaos

DOI

10.1142/S0218127425501706