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Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

  • *Corresponding author: Yaodan Huang

    *Corresponding author: Yaodan Huang 

The first author is supported by [the National Natural Science Foundation of China (No. 12301246) and Guangdong Provincial Natural Science Foundation (No. 2021A1515111004)]

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  • In [3] a mathematical model of the initiation and development of atherosclerosis involving LDL and HDL cholesterol, macrophages, and foam cells was introduced. The model is a highly nonlinear and coupled system of PDEs with a free boundary – the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions that bifurcate from the stationary annular solution in the combined longitude-latitude direction. After establishing various estimates for our PDE system, the Crandall-Rabinowitz theorem is applied to prove our main bifurcation theorem.

    Mathematics Subject Classification: Primary: 35B32, 35R35; Secondary: 35Q92.

    Citation:

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  • [1] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.
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    [3] A. FriedmanW. Hao and B. Hu, A free boundary problem for steady small plaques in the artery and their stability, Journal of Differential Equations, 259 (2015), 1227-1255.  doi: 10.1016/j.jde.2015.02.002.
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    [5] W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis: A mathematical model, PloS one, 9 (2014), e90497.
    [6] Y. Huang and B. Hu, Symmetry-breaking longitude bifurcations for a free boundary problem modeling small plaques in three dimensions, Journal of Mathematical Biology, 85 (2022), Paper No. 58, 44 pp. doi: 10.1007/s00285-022-01827-y.
    [7] C. McKay, S. McKee, N. Mottram, T. Mulholland, S. Wilson, S. Kennedy and R. Wadsworth, Towards a model of atherosclerosis, University of Strathclyde, 1983, 1-29.
    [8] D. MukherjeeL. Guin and S. Chakravarty, A reaction–diffusion mathematical model on mild atherosclerosis, Modeling Earth Systems and Environment, 5 (2019), 1853-1865. 
    [9] X. Zhao and B. Hu, Bifurcation for a free boundary problem modeling a small arterial plaque, Journal of Differential Equations, 288 (2021), 250-287.  doi: 10.1016/j.jde.2021.04.008.
    [10] X. Zhao and B. Hu, On the first bifurcation point for a free boundary problem modeling small arterial plaque, Mathematical Methods in Applied Sciences, 45 (2022), 4974-4988.  doi: 10.1002/mma.8087.
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