Convergence to separate variables solutions for a degenerate parabolic equation with gradient source

TL;DR

This paper studies the long-term behavior of solutions to a degenerate parabolic equation with a gradient-dependent source, demonstrating convergence to a unique profile and analyzing blowup phenomena using viscosity solutions and barrier functions.

Contribution

It introduces a novel analysis of the asymptotic behavior of solutions to a gradient-dependent degenerate diffusion equation, including convergence results and blowup analysis.

Findings

Solutions converge to a unique profile after rescaling.

Optimal decay rates and boundary estimates are established.

Blowup behavior of weak solutions is characterized.

Abstract

The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied.