Global solutions to the gradient flow equation of a nonconvex functional

We study the L2-gradient flow of the nonconvex functional F phi(u) := 1/2 integral((0,1)) phi(u(x)) dx, where phi(xi) := min(xi(2), 1). We show the existence of a global in time possibly discontinuous solution u starting from a mixed-type initial datum u(0), i. e., when u(0) is a piecewise smooth function having derivative taking values both in the region where phi'' > 0 and where phi'' = 0. We show that, in general, the region where the derivative of u takes values where phi'' = 0 progressively disappears while the region where phi'' is positive grows. We show this behavior with some numerical experiments.