This paper quantifies the speed of convergence and higherorder asymptotics of fast diffusion dynamics on $mathbf{R}^n$ to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by reexpressing 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 LaplaceBeltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finitedepth 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 Jochen Denzler
Jochen Denzler , University of Tennessee, Knoxville, TN, USA. Herbert Koch , Mathematisches Institut der Universitat Bonn, Germany. Robert J. McCann , University of Toronto, Ontario, Canada.Details Book
Author  :  Jochen Denzler 
Publisher  :  American Mathematical Society 
Data Published  :  30 March 2015 
ISBN  :  1470414082 
EAN  :  9781470414085 
Format Book  :  PDF, Epub, DOCx, TXT 
Number of Pages  :  81 pages 
Age +  :  15 years 
Language  :  English 
Rating  : 
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