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

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.