Long-time asymptotics for fully nonlinear homogeneous parabolic equations

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 $Φ^+$ and negative solution $Φ^-$, which satisfy the self-similarity relations \[ Φ^\pm (x,t) = λ^{α^\pm} Φ^\pm (λ^{1/2} x, λt). \] We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to $Φ^+$ ($Φ^-$) locally uniformly in $\R^n \times \R_+$. The anomalous exponents $α^+$ and $α^-$ are identified as the principal half-eigenvalues of a certain elliptic operator associated to $F$ in $\R^n$.