Title: Some qualitative properties for the Kirchhoff total variation flow Show Chinese title

Title: 某些关于Kirchhoff全变分流的定性性质

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_{\Omega}|Du|\right)\Delta_{1} u=0 & \text{in}\ & \Omega\times (0,+\infty) , \\ u=0 & \text{on} &\partial \Omega\times (0,+\infty),\\ u(x,0)=u_{0}(x) & \text{in} &\Omega , \end{array}\right. \end{equation*} where $\Omega\subset \mathbb{R}^{N}$ ($N\geq 1$) is a bounded smooth domain, $m :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ 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. Show Chinese abstract

Abstract: 本文我们关注的是以下涉及1-Laplace算子的Kirchhoff型问题：\begin{equation*} \left\{\begin{array}{llc} u_{t}-m\left(\int_{\Omega}|Du|\right)\Delta_{1} u=0 & \text{in}\ & \Omega\times (0,+\infty) , \\ u=0 & \text{on} &\partial \Omega\times (0,+\infty),\\ u(x,0)=u_{0}(x) & \text{in} &\Omega , \end{array}\right. \end{equation*}其中$\Omega\subset \mathbb{R}^{N}$($N\geq 1$) 是一个有界光滑区域，$m :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$是一个满足一些条件的递增连续函数，这些条件将在后面提及，而$\Delta_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$表示1-Laplace算子。这项工作的主要目的是从初始数据$u_{0}$和非线性函数$m$研究解在灭绝时间附近的存在性和渐近行为。