On determination of the bifurcation type for a free boundary problem modeling tumor growth
Xinyue Evelyn Zhao, Junping Shi
TL;DR
The paper addresses symmetry-breaking bifurcations in a two-dimensional free boundary tumor-growth model. It combines Crandall–Rabinowitz bifurcation theory with a pressure-decomposition technique and Fourier–Bessel expansions to locate nonradial bifurcation points $ mu_n= rac{n(n^2-1)}{R_S^3 M_n}$ from the radially symmetric branch, yielding nonradial branches with boundaries $r=R_S+ olinebreak[4] abla ext{cos}(n heta)$. A detailed second-order expansion shows all such bifurcations are pitchfork, with $ mu_n'(0)=0$ for every $n eq 1$, and numerical simulations near $ mu_2$ exhibit the two symmetric branches characteristic of pitchfork bifurcation; direction requires a third-order analysis. The results reveal a qualitative difference between the 2D and 3D versions of the same model and provide a rigorous framework for analyzing free-boundary bifurcations in tumor-growth models.
Abstract
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.