Higher-order time asymptotics of fast diffusion in Euclidean space. A dynamical systems approach.

Denzler, Jochen; Koch, Herbert; Mccann, Robert J.

ISBN 10: 1470414082 / ISBN 13: 9781470414085

Published by Providence, American Math. Soc, 2015

About this Item

Ex-library in GOOD condition with library-signature and stamp(s). Some traces of use. Ehem. Bibliotheksexemplar mit Signatur und Stempel. GUTER Zustand, ein paar Gebrauchsspuren. C-00006 9781470414085 Sprache: Englisch Gewicht in Gramm: 350.

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Bibliographic Details

• Title: Higher-order time asymptotics of fast diffusion in Euclidean space. A dynamical systems approach.
• Publisher: Providence, American Math. Soc
• Publication Date: 2015
• Language: English
• ISBN 10: 1470414082
• ISBN 13: 9781470414085
• Binding: Softcover
• Weight: 350 grams
• Seller catalogs:
  • U Physik

About this title

Synopsis

This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on $mathbfRn$ to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution becomes differentiable in Holder spaces on the cigar. The linearization of the dynamics is given by the Laplace-Beltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity.The authors provide a detailed study of the linear and nonlinear problems in Holder spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution, and higher order asymptotics. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

About the Author

Jochen Denzler, University of Tennessee, Knoxville, TN, USA.Herbert Koch, Mathematisches Institut der Universitat Bonn, Germany.Robert J. McCann, University of Toronto, Ontario, Canada.