We study the long-time asymptotics of solutions of the uniformly parabolic equation
$$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$
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
$$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$
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
$${\mathbb{R}^{n} \times \mathbb{R}_{+}}$$
. The anomalous exponents α+ and α− are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in
$${\mathbb{R}^{n}}$$
.