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Symmetry‐breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation
Author(s) -
Song Huijuan,
Wang Zejia,
Hu Baihui
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7008
Subject(s) - mathematics , bifurcation , uniqueness , symmetry breaking , boundary (topology) , monotone polygon , symmetry (geometry) , angiogenesis , mathematical analysis , boundary value problem , geometry , nonlinear system , physics , medicine , quantum mechanics
In this paper, we consider bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation. In which, the vasculature supplies nutrients to the tumor at a rate α , so that∂ σ ∂ n →+ α ( σ − σ ‾ ) = 0 holds on the boundary. For each α , we first establish the existence and uniqueness of radially symmetric stationary solutions, then prove that there exist a positive integer n ∗∗ and a sequence ( μ / γ ) n such that symmetry‐breaking stationary solutions bifurcate from the radially symmetric one for every ( μ / γ ) n (even n ≥ n ∗∗ ), where μ and γ denote the proliferation rate and the cell‐to‐cell adhesiveness, respectively. Particularly, for small α , we show that ( μ / γ ) 2 , ( μ / γ ) 4 , ( μ / γ ) 6 , ( μ / γ ) 8 , … are all bifurcation points, which includes the smallest bifurcation point that is the most significant one biologically; moreover, ( μ / γ ) n is monotone decreasing with respect to α for every n ≥ 2 , which implies that inhibiting angiogenesis has a positive impact in limiting the ability of the invasion of tumors, at least during very early stages of angiogenesis.