# 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