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Nonsymmetric traveling wave solution to a Hele-Shaw type tumor growth model

Yu Feng, Qingyou He, Jian-Guo Liu, Zhennan Zhou

TL;DR

This work studies boundary instabilities in a PME-derived Hele-Shaw tumor growth model by rigorously establishing the existence of nonsymmetric traveling wave solutions in a two-dimensional tube. The authors formulate a nonlinear map $F(\xi,\lambda)$ whose trivial branch corresponds to symmetric traveling waves and whose nontrivial zeros yield traveling-wave boundaries with shape $\xi(y)$ near $\varepsilon\cos(ly)$. Central to the analysis is the Fréchet derivative $F_\xi(0,\lambda)$, which reduces to an eigenvalue problem with eigenfunctions $\cos(ly)$ and eigenvalues $E(\lambda,l)$; zeros of $E(\lambda,l)$ pinpoint bifurcation points $\lambda_0^l$, enabling Crandall–Rabinowitz bifurcation to generate branches of nonsymmetric solutions. The main result shows that for each perturbation frequency $l\in\mathbb{N}$ there exists a local bifurcation branch $(\xi_{\varepsilon},\lambda_{\varepsilon})$ emanating from $(0,\lambda_0^l)$, confirming intrinsic boundary instability in the PME-derived Hele-Shaw tumor model. This provides a rigorous framework linking boundary morphology to nutrient consumption and supports the formation of finger-like protrusions in tumor growth scenarios.

Abstract

We consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. The model is derived by taking the incompressible limit of porous medium type equations, and the boundary instability of this model was recently studied in \cite{feng2022tumor} using asymptotic analysis. In this paper, we further prove the existence of nonsymmetric traveling wave solutions to the model in a two dimensional tube-like domain, which reflect intrinsic boundary instability in tumor growth dynamics.

Nonsymmetric traveling wave solution to a Hele-Shaw type tumor growth model

TL;DR

This work studies boundary instabilities in a PME-derived Hele-Shaw tumor growth model by rigorously establishing the existence of nonsymmetric traveling wave solutions in a two-dimensional tube. The authors formulate a nonlinear map whose trivial branch corresponds to symmetric traveling waves and whose nontrivial zeros yield traveling-wave boundaries with shape near . Central to the analysis is the Fréchet derivative , which reduces to an eigenvalue problem with eigenfunctions and eigenvalues ; zeros of pinpoint bifurcation points , enabling Crandall–Rabinowitz bifurcation to generate branches of nonsymmetric solutions. The main result shows that for each perturbation frequency there exists a local bifurcation branch emanating from , confirming intrinsic boundary instability in the PME-derived Hele-Shaw tumor model. This provides a rigorous framework linking boundary morphology to nutrient consumption and supports the formation of finger-like protrusions in tumor growth scenarios.

Abstract

We consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. The model is derived by taking the incompressible limit of porous medium type equations, and the boundary instability of this model was recently studied in \cite{feng2022tumor} using asymptotic analysis. In this paper, we further prove the existence of nonsymmetric traveling wave solutions to the model in a two dimensional tube-like domain, which reflect intrinsic boundary instability in tumor growth dynamics.
Paper Structure (14 sections, 9 theorems, 122 equations, 1 figure)

This paper contains 14 sections, 9 theorems, 122 equations, 1 figure.

Table of Contents

  1. introduction
  2. Model derivation
  3. A cell density model of PME type
  4. A formal derivation
  5. Bifurcation Analysis
  6. Model set up
  7. Symmetric solutions
  8. The perturbed system
  9. The linearized system
  10. Single mode perturbation
  11. A nonlinear functional and its Fréchet derivative.
  12. Existence of nontrivial bifurcation branches
  13. Incompressible limit
  14. Justification of the expansion

Key Result

Theorem 1.1

For each perturbation frequency $l\in \mathbb{N}$, there exists a bifurcation point $\lambda_0^l$ such that one can find a bifurcation branch $(\xi_{\epsilon}(y),\lambda_\epsilon)$ (parameterized by the small parameter $0<\epsilon\ll 1$) starting from $(0,\lambda_0^l)$ such that $F(\xi_\epsilon(y),\

Figures (1)

  • Figure 1: Fix $G_0\cdot c_B=100$. Left: plot for $E(\lambda,l)$ with $l=5$ and $\lambda\in[0,2000]$. Right: plot for $E(\lambda,l)$ with $\lambda=100$ and $l\in[0,20]$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Remark 3.5
  • Remark 3.6
  • proof
  • Theorem 3.7
  • Theorem 3.8
  • ...and 9 more