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

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 $t\rightarrow +\infty$.