Sharp growth rate for generalized solutions evolving by mean curvature plus a forcing term

When a hypersurface Sigma (t) evolves with normal velocity equal to its mea n curvature plus a forcing term g(x, t), the generalized (viscosity) soluti on may be "fattened" at some moment when Sigma (t) is singular. This phenom enon corresponds to nonuniqueness of codimension-one solutions. A specific type of geometric singularity occurs if Sigma (t) includes two smooth piece s, at the moment t = 0 when the two pieces touch each other. If each piece is strictly convex at that moment and at that point, then we show that fatt ening occurs at the rate t(1/3). That is, for small positive time, the gene ralized solution contains a ball of R-n of radius ct(1/3), but its compleme nt meets a ball of a larger radius kappa (0)t(1/3). In this sense, the shar p rate of fattening of the generalized solution is characterized. We assume that the smooth evolution of the two pieces of Sigma (t), considered separ ately, do not cross each other for small positive time.