Localization of Solutions of a Nonlinear Fokker-planck Equation with Dirichlet Boundary Conditions

u, = (0, for (x, f) E R x lR+, (1.1) W 0) = %(X) for x E IR, (1.2) where m > 1 and u0 is a given nonnegative function. It is well known [l, 11, 151 that if u. is continuous and bounded, problem (1. l), (1.2) has a unique nonnegative solution defined in a generalized sense, u(x, 1). Furthermore, if u,, is nontrivial and has compact support one can define the free boundary c(t) = sup(x E m: u(x, f) > 0) (1.3) for all t 2 0. This interface is continuous and monotonic increasing. Moreover,