### Abstract

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 ℝ^{n} × 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}.

Original language | English (US) |
---|---|

Pages (from-to) | 521-540 |

Number of pages | 20 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 38 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2010 |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*38*(3), 521-540. https://doi.org/10.1007/s00526-009-0297-3