UC Berkeley

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$u_t+F(D^{2}u)=0\mathrm{i}\mathrm{n}{\mathbb{R}}^n\times {\mathbb{R}}_{+},$$ 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 $${\mathrm{\Phi }}^{\pm }(x,t)={\lambda }^{{\alpha }^{\pm }}{\mathrm{\Phi }}^{\pm }({\lambda }^{1/2}x,\lambda t).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to $${\mathrm{\Phi }}^{+}$$ ( $${\mathrm{\Phi }}^{-}$$ ) 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 $${\mathbb{R}}^n$$ .