Global dynamics of a nonlocal non-uniformly parabolic equation arising from the curvature flow* *The first author is supported by NSF of China (No. 12271437 and No. 12071203). The second author is supported by NSF of China (No. 11971498) and NSF of Guangdong Province (No. 2019A1515011339). The third author is supported by NSF of China (No. 11871148).

Abstract

This paper studies a type of non-uniformly parabolic problem with nonlocal term ut=up(uxx+u−u¯)0<t<Tmax,0<x<a,ux(t,0)=ux(t,a)=00<t<Tmax,u(0,x)=u0(x)0<x<a, where p > 1, a > 0. First the classification of the finite-time blow-up/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. More importantly, we derive that any bounded solution converges to some steady state as t → +∞. The difficulties in proving this convergence result lie in the existence of a continuum of steady states and the lack of the comparison principle due to the introduction of nonlocal term. To conquer these difficulties, we combine the applications of Lojasiewicz–Simon inequality and energy estimates.