Long-time asymptotics for fully nonlinear homogeneous parabolic equations

TL;DR

This paper investigates the long-time behavior of solutions to fully nonlinear homogeneous parabolic equations, establishing the existence of self-similar solutions and their role as asymptotic limits for solutions with sign-definite initial data.

Contribution

It introduces a framework for analyzing long-time asymptotics of nonlinear parabolic equations, identifying unique self-similar solutions and their connection to principal eigenvalues.

Findings

Existence of unique positive and negative self-similar solutions.

Rescaled solutions converge to these self-similar solutions.

Identification of anomalous exponents as principal eigenvalues.

Abstract

We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the assumption that $g$ does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution $\Phi^+$ and negative solution $\Phi^-$, which satisfy the self-similarity relations \[ \Phi^\pm (x,t) = \lambda^{\alpha^\pm} \Phi^\pm (\lambda^{1/2} x, \lambda t). \] We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to $\Phi^+$ ($\Phi^-$) locally uniformly in $\R^n \times \R_+$. The anomalous exponents $\alpha^+$ and $\alpha^-$ are identified as the principal half-eigenvalues of a certain elliptic operator associated to $F$ in $\R^n$.