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 \({\Phi^+}\) (\({\Phi^-}\)) 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 \({\mathbb{R}^{n}}\) .
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Acknowledgements
The authors would like to express their appreciation to their thesis advisor, Lawrence C. Evans for his advice and guidance, and to thank the Department of Mathematics of UC Berkeley, for its support. We also thank Juan Luis Vázquez for his valuable comments and references, and Grigory Barenblatt for helpful comments. We are also indebted to an anonymous referee whose helpful comments greatly improved this article.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Armstrong, S.N., Trokhimtchouk, M. Long-time asymptotics for fully nonlinear homogeneous parabolic equations. Calc. Var. 38, 521–540 (2010). https://doi.org/10.1007/s00526-009-0297-3
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DOI: https://doi.org/10.1007/s00526-009-0297-3