TITLE

# Long-time asymptotics for fully nonlinear homogeneous parabolic equations

AUTHOR(S)
Armstrong, Scott N.; Trokhimtchouk, Maxim
PUB. DATE
July 2010
SOURCE
Calculus of Variations & Partial Differential Equations;Jul2010, Vol. 38 Issue 3/4, p521
SOURCE TYPE
DOC. TYPE
Article
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 Î¦+ and negative solution Î¦âˆ’, 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 $${\Phi^+}$$ ( $${\Phi^-}$$) locally uniformly in $${\mathbb{R}^{n} \times \mathbb{R}_{+}}$$ . The anomalous exponents Î±+ and Î±âˆ’ are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in $${\mathbb{R}^{n}}$$ .
ACCESSION #
50354929

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