Evolution of nonparametric surfaces with speed depending on curvature function (Q2704333)

In recent work the authors [Northeast. Math. J. 15, No. 3, 301-314 (1999)] proved the existence of classical solutions of fully nonlinear, nonuniformly parabolic equations of the form NEWLINE\[NEWLINE -D_tu = {{\psi(x,t)}\over{f(\kappa(u))}} \quad\text{in}\quad \Omega\times(0,\infty) \leqno(*) NEWLINE\]NEWLINE subject to the first initial-boundary condition. Here \(\Omega\) is a smooth strictly convex subdomain of \({\mathbb R}^n\), \(\kappa(u)\) denotes the principal curvatures \(\kappa_1,\dots,\kappa_n\) of the graph of \(u\), \(\psi\) is a positive function, and \(f\) is a symmetric positive function satisfying a number of conditions. A typical example is \(f=(S_k)^{1/k}\), where for any integer between \(1\) and \(n\), \(S_k\) is the \(k\)-th elementary symmetric function. NEWLINENEWLINENEWLINEHere the authors investigate the asymptotic behaviour of these solutions as \(t\rightarrow\infty\). Under suitable assumptions on the data they show that \(u(\cdot,t)\) converges to a solution \(w\) of an elliptic equation closely related to \((*)\). In the radially symmetric case \(w\) is a portion of a hemisphere.