Long-time asymptotics for fully nonlinear homogeneous parabolic equations

Scott N. Armstrong, Maxim Trokhimtchouk

Research output: Contribution to journalArticle

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+-) 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 languageEnglish (US)
Pages (from-to)521-540
Number of pages20
JournalCalculus of Variations and Partial Differential Equations
Volume38
Issue number3
DOIs
StatePublished - Jul 1 2010

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ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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