Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

TL;DR

This paper proves the long-term convergence of solutions to a degenerate parabolic equation with a nonlocal gradient term, identifying the limit as the stationary solution when initial mass equals one, and distinguishes this behavior from other initial masses.

Contribution

It establishes the convergence of positive solutions with unit initial mass to a stationary state and introduces a novel analysis based on a monotonicity property and constrained minimization.

Findings

Solutions with initial mass one converge to a stationary solution.

Different initial masses lead to zero or blow-up behavior.

Monotonicity of the gradient norm is key to the analysis.

Abstract

We prove convergence of positive solutions to \[ u_t = u\Delta u + u\int_{\Omega} |\nabla u|^2, \qquad u\rvert_{\partial\Omega} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $\Omega\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_\Omega u_0=1$ and identify the $W_0^{1,2}(\Omega)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_\Omega u_0<1$ and $\int_\Omega u_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_{\Omega} |\nabla u|^2$ along trajectories and the analysis of an associated constrained minimization problem. Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviour