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

TL;DR

This paper analyzes a free boundary problem modeling tumor growth, revealing that all symmetry-breaking bifurcations in the 2D case are pitchfork bifurcations, and highlights differences with 3D cases.

Contribution

It provides a detailed bifurcation analysis of a tumor growth model, establishing the nature of symmetry-breaking bifurcations in 2D using high-order expansions.

Findings

All symmetry-breaking bifurcations are pitchfork bifurcations.

Differences between 2D and 3D bifurcation behaviors.

Application of Crandall-Rabinowitz theorem to free boundary problems.

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.