Global solutions to the gradient flow equation of a nonconvex functional

Abstract. We study the L2-gradient flow of the nonconvex functional F(u):=∫(0,1)f(ux)dx, where f(ξ):=min(ξ2,1). We show the existence of a global in time possibly discontinuous solution starting from a mixed-type initial datum, i.e., when a piecewise smooth function having derivative taking values both in the region where the second derivative of f is strictly positive (so called "good region") and where it is zero (so called "bad region"). We show that, in general, the bad region progressively disappears while the good region grows. We show this behaviour with numerical experiments.