Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems

We study the fully nonlinear parabolic equation F(D(2)u(m)) - u(t) = 0 in Omega x (0, +infinity), m >= 1, with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Omega subset of R-n, provided that the operator F is uniformly elliptic and positively homogeneous of order one. We prove that the renormalized limit of parabolic flow u(x, t) as t -> +infinity is the corresponding positive eigenfunction which solves F(D-2 phi)+mu phi(P) = 0(.) in Omega, where 0 < p := 1/m <= 1 and mu > 0 is the corresponding eigenvalue. We also show that some geometric property of the positive initial data is preserved by the parabolic flow, under the additional assumptions that Omega is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, log(phi) is concave in the case p = 1, and phi(1-p/2) is concave for 0< p <1. (C) 2013 Elsevier Inc. All rights reserved.