Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics

Wasserstein Metric and Large–time Asymptotics of Nonlinear Diffusion Equations

We review here various recent applications of Wassertein–type metrics to both nonlinear partial differential equations and integro–differential equations. Among others, we can describe the asymptotic behavior of nonlinear friction equations arising in the kinetic modelling of granular flows, and the growth of the support in nonlinear diffusion equations of porous medium type. Further examples i...

Fast diffusion equations: matching large time asymptotics by relative entropy methods

A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the...

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

Fast crack propagation by surface diffusion.

We present a continuum theory which describes the fast growth of a crack by surface diffusion. This mechanism overcomes the usual cusp singularity by a self-consistent selection of the crack tip radius. It predicts the saturation of the steady state crack velocity appreciably below the Rayleigh speed and tip blunting. Furthermore, it includes the possibility of a tip splitting instability for h...

Long Time Asymptotics for Fast Diffusion via Dynamical Systems Methods

For the fast diffusion equation in the mass preserving parameter range, we obtain sharp asymptotic convergence rates to the Barenblatt solution with respect to the relative L∞ norm from spectral gaps by establishing a nonlinear differentiable semiflow in Hölder spaces on a Riemannian manifold called cigar manifold. On this manifold, the equation becomes uniformly parabolic. It is possible to ob...