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

TL;DR

This paper proves the existence and stability of solutions for a neurologically-inspired parabolic PDE with nonlinear, delayed boundary conditions, modeling protein production with feedback mechanisms and analyzing bifurcation behavior.

Contribution

It introduces a novel PDE model with nonlinear, time-delayed boundary conditions and establishes conditions for solution existence, stability, and bifurcation analysis.

Findings

Existence of positive, bounded solutions established.

Conditions for stability and Hopf bifurcation derived.

Model captures feedback regulation in protein production.

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.