Avner Friedman Mathematical Biosciences Institute

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Presentation transcript:

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

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

tumor Ω(t)

tumor Ω(t)

unstable? stable? stable unstable unstable? stable?

linearly unstable linearly unstable stable linearly stable linearly unstable Hopf bifurcation stable

Tumor in fluid-like tissue

Model Equations

Symmetry-breaking bifurcations

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

Loss of Stability for Stokes Equation

Loss of stability for Darcy’s law M2 M2

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