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Linearized Eigenvalues for a Free Boundary Problem Modeling Two-phase Tumor Growth by Cui, Shangbin; Zheng, Jiayue is a scholarly article available to read on EtoBox.

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In this paper we study a linearized eigenvalue problem derived from a a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. The reduced form of this eigenvalue problem is a $2$-system of a first-order nonlocal singular differential-integral equation in a ball coupled by a third-order elliptic pseudo-differential equation in the unit sphere. The singularity joined with non-localness of the first-order equation causes the main difficulty of this problem. By using Fourier expansion via a basis of spherical harmonic functions and some techniques for solving singular differential integral equations developed in some previous literature, we prove that there exists a null sequence $\{\gamma_k\}_{k=2}^{\infty}$ for the surface tension coefficient $\gamma$, with each of them being an eigenvalue of the linearized problem, i.e., if $\gamma=\gamma_k$ for some $k\geq 2$ then the linearized problem has extra nontrivial solutions besides the standard nontrivial solutions, and if $\gamma\neq\gamma_k$ for all $k\geq 2$ then the linearized problem does not have other nontrivial solutions than the standard nontrivial solutions

Author

Cui, Shangbin; Zheng, Jiayue

Published

2013

Language

EN