Asymptotics Near Extinction for Nonlinear Fast Diffusion on a Bounded Domain.

Area of Science:

• Partial Differential Equations
• Mathematical Physics
• Nonlinear Dynamics

Background:

• Fast diffusion equations on bounded domains with vanishing boundary trace exhibit finite-time extinction.
• The vanishing profile is determined by the initial data, but convergence rates are not fully understood.

Purpose of the Study:

• To quantify the rate of convergence to the vanishing profile in rescaled variables, uniformly in relative error.
• To analyze the influence of non-integrable zero modes on the convergence rate.
• To refine and confirm existing conjectures regarding nonlinear dynamics and eigenmode approximation.

Main Methods:

• Analysis of Sobolev-subcritical fast diffusion equations.
• Rescaling of variables to study convergence rates.
• Investigation of spectral properties and zero modes.
• Development of a new, simpler analytical approach.

Main Results:

• Convergence rates are either exponentially fast (linked to spectral gap) or algebraically slow (in the presence of non-integrable zero modes).
• Nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to twice the spectral gap.
• A new method accommodates zero modes, improving on prior results.

Conclusions:

• The study provides a comprehensive understanding of extinction dynamics in fast diffusion.
• It confirms and refines a 1980 conjecture by Berryman and Holland.
• The developed approach offers a more robust analysis, particularly in cases with non-isolated vanishing profiles.