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Critical exponents for large time asymptotics for fractal Hamilton‐Jacobi‐KPZ equations
Author(s) -
Karch G.,
Woyczynski W.A.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700865
Subject(s) - brownian motion , bounded function , nonlinear system , fractal , mathematics , generator (circuit theory) , infinitesimal , mathematical proof , operator (biology) , mathematical analysis , space (punctuation) , diffusion , statistical physics , physics , geometry , quantum mechanics , computer science , power (physics) , statistics , biochemistry , chemistry , repressor , transcription factor , gene , operating system
Nonlinear and nonlocal evolution equations of the form u t = ℒ u ± |∇ u | q , where ℒ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived (see, [6]) as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this note is to report properties of solutions to this equation resulting from the interplay between the strengths of the “diffusive” linear and “hyperbolic” nonlinear terms, posed in the whole space R N , and supplemented with nonnegative, bounded, and sufficiently regular initial conditions. The full text of the paper, including complete proofs and other results, will appear in the Transactions of the American Mathematical Society. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)