A Free Boundary Problem from Stochastic Control - Adam Oberman Math

Problem Formulation

Consider the fully nonlinear convex Partial Differential Equation

u t = m a x (2 u x x, u x x)

in the interval

Ω = [ - 1,1]

with homogeneous Dirichlet boundary conditions

$u = 0 \text{ on } \partial \Omega$ .

There is a free boundary where the maximum switches, located at

x (t) = {u x x (x, t) = 0}

Take, for example, initial data

u (x,0) = sin(π x)

Then a plot of the solution and the location of the free boundary is below. Notice the free boundary moves briefly to the right, before moving to the left.

Plots

Question

Is there a formula for the speed of the free boundary?