Higher-order time asymptotics of fast diffusion in Euclidean space : a dynamical systems approach
著者
書誌事項
Higher-order time asymptotics of fast diffusion in Euclidean space : a dynamical systems approach
（Memoirs of the American Mathematical Society, no. 1101）
American Mathematical Society, 2015, c2014
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注記
"Volume 234, number 1101 (first of 5 numbers), March 2015"
Includes bibliographical references (p. 79-81)
内容説明・目次
内容説明
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 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.
目次
Introduction
Overview of obstructions and strategies, and notation
The nonlinear and linear equations in cigar coordinates
The cigar as a Riemannian manifold
Uniform manifolds and Holder spaces
Schauder estimates for the heat equation
Quantitative global well-posedness of the linear and nonlinear equations in Holder spaces
The spectrum of the linearized equation
Proof of Theorem 1.1
Asymptotic estimates in weighted spaces: The case $m< \frac{n}{n+2}$
Higher asymptotics in weighted spaces: The case $m> \frac{n}{n+2}$.
Proof of Theorem 1.2 and its corollaries
Appendix A. Pedestrian derivation of all Schauder estimates
Bibliography
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