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.
|Original language||English (US)|
|Number of pages||20|
|Journal||Calculus of Variations and Partial Differential Equations|
|State||Published - Jul 2010|
ASJC Scopus subject areas
- Applied Mathematics