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Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach. (English) Zbl 1315.35004
Mem. Am. Math. Soc. 1101, v, 81 p. (2015).
Summary: This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on \(\mathbb R^n\) 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.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
76S05 Flows in porous media; filtration; seepage
35C06 Self-similar solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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