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<p>In this paper, we study a nonlinear free boundary problem on the radial growth of a two-layer solid tumor with a quiescent core. The tumor surface and its inner interface separating the proliferating cells and the quiescent cells are both free boundaries. By deeply analyzing their relationship and employing the maximum principle, we show that this problem is globally well-posed and prove the existence of a unique positive threshold <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10198_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma ^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>σ</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> such that the problem admits a unique stationary solution with a quiescent core if and only if the externally supplied nutrient <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10198_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\({\bar{\sigma }}&gt; \sigma ^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>&gt;</mo> <msup> <mi>σ</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. The stationary solution is globally asymptotically stable. The formation of the quiescent core and its interesting connection with the necrotic core are also given.</p>

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Analysis of a Nonlinear Free Boundary Problem Modeling the Radial Growth of Two-Layer Tumors

  • 期刊论文

  • 发表日期：

    2025年8月12日

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  • Junde Wu,
  • Hao Xu,
  • Yuehong Zhuang

摘要

In this paper, we study a nonlinear free boundary problem on the radial growth of a two-layer solid tumor with a quiescent core. The tumor surface and its inner interface separating the proliferating cells and the quiescent cells are both free boundaries. By deeply analyzing their relationship and employing the maximum principle, we show that this problem is globally well-posed and prove the existence of a unique positive threshold \(\sigma ^*\) σ ∗ such that the problem admits a unique stationary solution with a quiescent core if and only if the externally supplied nutrient \({\bar{\sigma }}> \sigma ^*\) σ ¯ > σ ∗ . The stationary solution is globally asymptotically stable. The formation of the quiescent core and its interesting connection with the necrotic core are also given.