Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model

In this article, we consider a free boundary problem modeling the growth of a double-layered tumor, which contains quiescent cells and proliferating cells. Compared with that composed of necrotic cells and proliferating cells, the remarkable feature of this model is that quiescent cells actively consume nutrients, introducing an additional nutrient decay term, but at a rate different from that of proliferating cells. The parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> μ in model represents the “aggressiveness” of the tumor. A threshold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msub> </m:math> {μ }* is determined such that the radially symmetric stationary solution is linearly stable for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msub> </m:math> μ < {μ }* and unstable for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo>></m:mo> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msub> </m:math> μ > {μ }* under non-radially symmetric perturbations. Moreover, it is shown that there exist a sequence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>≥</m:mo> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \{{μ }ₙ\}(≥ {μ }*) and a positive integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> {n}* *≥ 2 , such that a branch of symmetry-breaking stationary solutions bifurcates from the radially symmetric stationary solution for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">even</m:mi> <m:mspace width="0.33em"/> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:msup> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> μ ={μ }ₙ({even}{0.33em}n≥ {n}* *) .