Some qualitative properties for the Kirchhoff total variation flow

Abstract

In this paper we are concerned with the following Kirchhoff type problem involving the 1-Laplace operator : \begin{equation*} \left\{\begin{array}{llc} u_{t}-m\left(\int_Ω|Du|\right)Δ_{1} u=0 & \text{in}\ & Ω\times (0,+\infty) , \\ u=0 & \text{on} &\partial Ω\times (0,+\infty),\\ u(x,0)=u_{0}(x) & \text{in} &Ω, \end{array}\right. \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ ($N\ge 1$) is a bounded smooth domain, $m:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is an increasing continuous function that satisfies some conditions which will be mentioned further down, and ${\Delta }_{1}u=\text{div}\left(\frac{Du}{|Du|}\right)$ denotes the 1-Laplace operator. The main purpose of this work is to investigate from the initial data $u_0$ and the nonlinear function $m$ the existence and asymptotic behavior of solutions near the extinction time.