Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation, 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, 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 ℝⁿ × ℝ₊. The anomalous exponents α⁺ and α^- are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ℝⁿ.