On the stability and asymptotic behavior for a quasi-linear parabolic flow

Abstract

In this paper, we study the stability property and asymptotic behavior for a quasi-linear parabolic flow in the whole line. We first show the existence and uniqueness of global solutions of the problem. Then we study the stability of the solution to the straight line. We prove the asymptotic behavior or the convergence of the global solution. Similar to the behavior of solutions to heat equation, we prove that the stationary line attracts the graphical curves which surround it.