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arXiv · 2507.09783

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

Gangadhara Boregowda·Michael R. Lindstrom
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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 $\tau \geq 0$ 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 non-trivial steady-state solution and provide sufficient conditions on the nonlinearity parameter, boundary flux, and time delay that ensure the occurrence of a Hopf bifurcation.

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Gangadhara Boregowda, Michael R. Lindstrom. 2025-07-13. Existence and Stability Theory of a Neurologically-Inspired Parabolic PDE Model with a Nonlinear Time-Delayed Boundary Condition. https://arxiv.org/abs/2507.09783

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