Global dynamics of a parabolic type equation arising from the curvature flow
This paper studies a type of degenerate parabolic problem with nonlocal term \beginequation* \begincases u_t=u^p(u_xx+u-\baru) & 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, \endcases \endequation* where , . 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 .