**Memoirs of the American Mathematical Society**

2014;
81 pp;
Softcover

MSC: Primary 35;
Secondary 33; 37; 58; 76

Print ISBN: 978-1-4704-1408-5

Product Code: MEMO/234/1101

List Price: $70.00

Individual Member Price: $42.00

**Electronic ISBN: 978-1-4704-2028-4
Product Code: MEMO/234/1101.E**

List Price: $70.00

Individual Member Price: $42.00

# Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach

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*Jochen Denzler; Herbert Koch; Robert J. McCann*

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.

The authors 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.

#### Table of Contents

# Table of Contents

## Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Overview of Obstructions and Strategies, and Notation 1118 free
- Chapter 3. The nonlinear and linear equations in cigar coordinates 1522
- Chapter 4. The cigar as a Riemannian manifold 1926
- Chapter 5. Uniform manifolds and Hölder spaces 2128
- Chapter 6. Schauder estimates for the heat equation 2936
- Chapter 7. Quantitative global well-posedness of the linear and nonlinear equations in Hölder spaces 3542
- Chapter 8. The spectrum of the linearized equation 4754
- Chapter 9. Proof of Theorem 1.1 5966
- Chapter 10. Asymptotic estimates in weighted spaces: The case 𝑚<𝑛/(𝑛+2) 6370
- Chapter 11. Higher asymptotics in weighted spaces: The case 𝑚>𝑛/(𝑛+2). Proof of Theorem 1.2 and its corollaries. 6572
- Appendix A. Pedestrian derivation of all Schauder Estimates 7380
- Bibliography 7986
- Back Cover Back Cover194