1. Avner Friedman Mathematical Biosciences Institute
Symmetry-breaking bifurcations of free-boundary problems arising in cancer models
Avner Friedman
Mathematical Biosciences Institute

2. V= velocity of the free boundary
Hele-Shaw problem
Free boundary
= mean curvature
V= velocity of the free boundary

3. tumor Ω(t)

4. tumor Ω(t)

10. unstable?
stable?
stable
unstable
unstable?
stable?

12. linearly
unstable
linearly
unstable
stable
linearly
stable
linearly
unstable
Hopf
bifurcation
stable

14. Tumor in fluid-like tissue

15. Model Equations

17. Symmetry-breaking bifurcations

18. For Darcy’s Law μ γ M2 M3 M4 M5 μ γ
Mn1
Mn2
Mn3
Mn4

19. Loss of Stability for Stokes Equation

20. Loss of stability for Darcy’s lawM2 M2

22. Questions How far can the bifurcaion branches be continued?
More general tumor models (several types
of cells): are there stationary spherical solutions? Stability and bifurcation.
Global stability of the spherical solutions