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| In this talk we consder a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. 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. |
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