Initial Motion of the Free Boundary for a Non-linear Diffusion Equation

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Abstract
The free boundary between v > 0 and v = 0 for the porous medium equation v1= |∇v|2 +nv∇2v can remain stationary for some positive waiting-time and then start moving. It is of interest to know the way in which any part of the boundary first moves. It is already known that if the waiting time is given by purely local considerations then the boundary speed is continuous, i.e. the initial speed is zero. For cases where the boundary moves before the time found by simply considering v(x, 0) close to the boundary, a perturbation analysis indicates that it starts moving with positive speed. Two-dimensional problems show the possible formation of vertices in the boundary. At these points the normal velocity jumps from zero to some positive value.

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