Global dynamics of a parabolic type equation arising from the curvature flow

TL;DR

This paper analyzes a degenerate parabolic equation with a nonlocal term related to curvature flow, classifies blowup and global existence, and proves convergence of solutions to steady states using energy methods.

Contribution

It provides a comprehensive classification of finite-time blowup and global solutions, and explicitly characterizes all steady states for the equation.

Findings

Classification of blowup and global solutions based on energy functional

Explicit expressions for all nonnegative steady states

Proof of convergence of bounded solutions to steady states

Abstract

This paper studies a type of degenerate parabolic problem with nonlocal term \begin{equation*} \begin{cases} u_t=u^p(u_{xx}+u-\bar{u}) & 0<t<T_{{\max}},\ 0<x<a, u_x(0,t)=u_x(a,t)=0 & 0<t<T_{{\max}}, u(x,0)=u_0(x) & 0<x<a, \end{cases} \end{equation*} where $p>1$, $a>0$. In this paper, the classification of the finite-time blowup/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. Moreover, we combine the applications of Lojasiewicz-Simon inequality and energy estimates to derive that any bounded solution with positive initial data converges to some steady state as $t\rightarrow +\infty$.