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Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach
About this Title
Jochen Denzler, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996, Herbert Koch, Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany and Robert J. McCann, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 234, Number 1101
ISBNs: 978-1-4704-1408-5 (print); 978-1-4704-2028-4 (online)
DOI: https://doi.org/10.1090/memo/1101
Published electronically: June 4, 2014
MSC: Primary 35B40; Secondary 33C50, 35K61, 37L10, 58J50, 76S05
Table of Contents
Chapters
- 1. Introduction
- 2. Overview of Obstructions and Strategies, and Notation
- 3. The nonlinear and linear equations in cigar coordinates
- 4. The cigar as a Riemannian manifold
- 5. Uniform manifolds and Hölder spaces
- 6. Schauder estimates for the heat equation
- 7. Quantitative global well-posedness of the linear and nonlinear equations in Hölder spaces
- 8. The spectrum of the linearized equation
- 9. Proof of Theorem
- 10. Asymptotic estimates in weighted spaces: The case $m< \frac {n}{n+2}$
- 11. Higher asymptotics in weighted spaces: The case $m> \frac {n}{n+2}$. Proof of Theorem and its corollaries.
- A. Pedestrian derivation of all Schauder Estimates
Abstract
This paper quantifies the speed of convergence and higher-order 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 re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution becomes differentiable in Hölder 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. We provide a detailed study of the linear and nonlinear problems in Hölder 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.- Sigurd Angenent, Large time asymptotics for the porous media equation, Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 21–34. MR 956056, DOI 10.1007/978-1-4613-9605-5_{2}
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