## Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach (Memoirs of the American Mathematical Society)

Author(s): Jochen Denzler

Date: Format: pdf Language: English ISBN/ASIN: 1470414082
Pages: OCR: Quality: ISBN13:
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.