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

作者：Kim Soojung^*; Lee Ki Ahm

来源：Journal of Differential Equations, 2013, 254(8): 3259-3306.

DOI：10.1016/j.jde.2013.01.015

• Fully nonlinear parabolic equation
• Large time behavior
• Heat equation
• Porous medium equation
• Fully nonlinear elliptic eigenvalue problem

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.

• 出版日期2013-4-15