Sharp growth rate for generalized solutions evolving by mean curvature plus a forcing term (Q2744700)

In work of \textit{G.~Barles, H. M.~Soner} and \textit{P. E. Souganidis} [SIAM J. Control. Optim. 31, 439--469 (1993; Zbl 0785.35049)] and \textit{G.~Bellettini} and \textit{M.~Paolini} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (9) Rend. Lincei Mat. Appl. 5, 229--236 (1994; Zbl 0826.35051)] it was observed that for a hypersurface \(\Sigma(t)\subset \mathbb{R}^{n+1}\) evolving with normal velocity equal to its mean curvature plus a forcing term \(g(x,t)\), the generalized (viscosity) solution may ``fatten'' at some time when \(\Sigma(t)\) develops a singularity. This phenomenon corresponds to nonuniqueness of codimension one solutions.NEWLINENEWLINEA specific type of geometric singularity is the following. Two smooth, separately evolving hypersurfaces \(\Sigma^{\pm}(t)\) are disjoint for \(t<0\); they touch at a single point \(x_0\) when \(t=0\), and then again become disjoint for \(t>0\). Assuming that each piece is strictly convex at the point of touching, the authors show that fattening occurs at the rate \(t^{1/3}\). More precisely, they show that there exist positive constants \(c,\kappa_0\) and \(\delta\), such that for each \(t\in(0,\delta)\), the generalized solution at time \(t\) contains a ball of radius \(ct^{1/3}\), and its complement intersects a ball of larger radius \(\kappa_0t^{1/3}\). This result improves the lower bound \(ct^{1/2}\) for the rate of fattening obtained by the second author [\textit{Y. Koo}, Commun. Partial Differ. Equ. 24, 1035--1053 (1999; Zbl 0935.35035)].