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

Abstract

We study the fully nonlinear parabolic equationF(D2um)−ut=0in Ω×(0,+∞),m⩾1, with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Ω⊂Rn, 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→+∞ is the corresponding positive eigenfunction which solvesF(D2φ)+μφp=0in Ω, where 0<p:=1m⩽1 and μ>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 Ω is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, log(φ) is concave in the case p=1, and φ1−p2 is concave for 0<p<1.