TY - JOUR
T1 - Continuation along bifurcation branches for a tumor model with a necrotic core
AU - Hao, Wenrui
AU - Hauenstein, Jonathan D.
AU - Hu, Bei
AU - Liu, Yuan
AU - Sommese, Andrew J.
AU - Zhang, Yong Tao
N1 - Funding Information:
Acknowledgements W. Hao was supported by the Dunces Chair of the University of Notre Dame and NSF grant DMS-0712910. J.D. Hauenstein was supported by Texas A&M University and NSF grant DMS-0915211. A.J. Sommese was supported by the Dunces Chair of the University of Notre Dame and NSF grant DMS-0712910. Y.-T. Zhang was partially supported by NSF grant DMS-0810413.
PY - 2012/11
Y1 - 2012/11
N2 - We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0 < ρ < R, there exists a radially-symmetric stationary solution with tumor free boundary r = R and necrotic free boundary r = ρ. The system depends on a positive parameter μ, which describes tumor aggressiveness, and for a sequence of values μ2 < μ3 <. . . , there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor μ. By continuously changing μ using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.
AB - We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0 < ρ < R, there exists a radially-symmetric stationary solution with tumor free boundary r = R and necrotic free boundary r = ρ. The system depends on a positive parameter μ, which describes tumor aggressiveness, and for a sequence of values μ2 < μ3 <. . . , there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor μ. By continuously changing μ using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.
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U2 - 10.1007/s10915-012-9575-x
DO - 10.1007/s10915-012-9575-x
M3 - Article
AN - SCOPUS:84869861936
SN - 0885-7474
VL - 53
SP - 395
EP - 413
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 2
ER -