Stability of periodic solution for a free boundary problem modeling small plaques

Mathematical models describing the growth of plaque in the arteries (e.g., Friedman and Hao (2015), Friedman et al. (2015), Hao and Friedman (2014), McKay et al. (2005) and Mukherjee et al. (2019)) were introduced. All of these models include the interaction of the "bad" cholesterols, low-density lipoprotein (LDL), and the "good" cholesterols, high-density lipoprotein (HDL), in triggering whether plaque will grow or shrink. Because the blood vessels tend to be circular, 2D cross-section model is a good approximation, and the 2D models are studied in Friedman et al. (2015), Zhang et al. (2023) and Zhao and Hu (2022). A bifurcation into a 3D plaque was recently studied in Huang and Hu (2022). All of these models assume a constant supply of LDL and HDL from the blood vessel. In reality, nutrient concentration changes with the intake of food, which happens very often in a periodic manner. When the LDL and HDL supplies from the blood vessel are periodic and are not too far away from the prevalent values, a periodic solution was obtained in Huang and Hu (2023). In this paper, we carry out the linear stability analysis of this periodic solution and provide simulation results to confirm our analysis.