arXiv subjects

Xinyue Evelyn Zhao

Publications and source records attributed to Xinyue Evelyn Zhao.

8 recordsLinked to original sources

arXiv2025

On determination of the bifurcation type for a free boundary problem modeling tumor growth

Many mathematical models in different disciplines involve the formulation of free boundary problems, where the domain boundaries are not predefined. These models present unique challenges, notably the nonlinear coupling between the solution and the boundary, which complicates the identification of bifurcation types. This paper mainly investigates the structure of symmetry-breaking bifurcations in a two-dimensional free boundary problem modeling tumor growth. By expanding the solution to a high order with respect to a small parameter and computing the bifurcation direction at each bifurcation point, we demonstrate that all the symmetry-breaking bifurcations occurred in the model, as established by the Crandall-Rabinowitz Bifurcation From Simple Eigenvalue Theorem, are pitchfork bifurcations. These findings reveal distinct behaviors between the two-dimensional and three-dimensional cases of the same model.

math.AP↗

arXiv2024

Optimal control of treatment in a free boundary problem modeling multilayered tumor growth

We study the optimal control problem of a free boundary PDE model describing the growth of multilayered tumor tissue in vitro. We seek the optimal amount of tumor growth inhibitor that simultaneously minimizes the thickness of the tumor tissue and mitigates side effects. The existence of an optimal control is established, and the uniqueness and characterization of the optimal control are investigated. Numerical simulations are presented for some scenarios, including the steady-state and parabolic cases.

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arXiv2022

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

Phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, we develop a new analytical approach based on bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: Allen-Cahn equation, Cahn-Hilliard equation, and Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

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arXiv2021

Two neural-network-based methods for solving obstacle problems

Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization of the obstacle problems. We rigorously prove the convergence of the two schemes and derive the convergence rates with the number of neurons $N$. In the simulations, we use two example problems (1-D & 2-D) to verify the convergence rate of the methods and the quality of the results.

math.NA↗

arXiv2020

Convergence analysis of neural networks for solving a free boundary system

Free boundary problems deal with systems of partial differential equations, where the domain boundaries are apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problems either theoretically or numerically. In this paper, we develop a novel approach for solving a modified Hele-Shaw problem based on neural network discretization. The existence of the numerical solution with this discretization is established theoretically. We also numerically verify this approach by computing the symmetry-breaking solutions which are guided by the bifurcation analysis near the radially-symmetric branch. Moreover, we further verify the capability of this approach by computing some non-radially symmetric solutions which are not characterized by any theorems.

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arXiv2020

On the first bifurcation point for a free boundary problem modeling small arterial plaque

Atherosclerosis occurs when plaque clogs the arteries. It is a leading cause of death in the United States and worldwide. In this paper, we study the bifurcation for a highly nonlinear and highly coupled PDE model of plaque formation in the early stage of atherosclerosis. The model involves LDL and HDL cholesterols, macrophage cells as well as foam cells, with the interface separating the plaque and blood flow region being a free boundary. We establish the first bifurcation point for the system corresponding to $n=1$ mode. The symmetry-breaking stationary solution studied in this paper might be helpful in understanding why there exists arterial plaque that is often accumulated more on one side of the artery than the other.

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arXiv2020

Bifurcation for a free boundary problem modeling a small arterial plaque

Atherosclerosis, hardening of the arteries, originates from small plaque in the arteries; it is a major cause of disability and premature death in the United States and worldwide. In this paper, we study the bifurcation of a highly nonlinear and highly coupled PDE model describing the growth of arterial plaque in the early stage of atherosclerosis. The model involves LDL and HDL cholesterols, macrophage cells as well as foam cells, with the interface separating the plaque and blood flow regions being a free boundary. We establish finite branches of symmetry-breaking stationary solutions which bifurcate from the radially symmetric solution. Since plaque in reality is unlikely to be strictly radially symmetric, our result would be useful to explain the asymmetric shapes of plaque.

math.AP↗

arXiv2019

The Impact of Time Delay in a Tumor Model

In this paper we consider a free boundary tumor growth model with a time delay in cell proliferation and study how time delay affects the stability and the size of the tumor. The model is a coupled system of an elliptic equation, a parabolic equation and an ordinary differential equation. It incorporates the cell location under the presence of time delay, with the tumor boundary as a free boundary. A parameter $μ$ in the model is proportional to the "aggressiveness" of the tumor. It is proved that there exists a unique classical radially symmetric stationary solution $(σ_*, p_*, R_*)$ which is stable for any $μ> 0$ with respect to all radially symmetric perturbations (c.f. \cite{delay1}). However, under non-radially symmetric perturbations, we prove that there exists a critical value $μ_*$, such that if $μ<μ_*$ then the stationary solution $(σ_*, p_*, R_*)$ is linearly stable; whereas if $μ>μ_*$ the stationary solution is unstable. It is actually unrealistic to expect the problem to be stable for large tumor aggressiveness parameter, therefore our result is more reasonable. Furthermore, we established that adding the time delay in the model would result in a larger stationary tumor, and if the tumor aggressiveness parameter is larger, then the time delay would have a greater impact on the size of the tumor.

math.AP↗