Some qualitative properties for the Kirchhoff total variation flow

TL;DR

This paper studies a Kirchhoff-type total variation flow involving the 1-Laplace operator, focusing on existence and long-term behavior of solutions depending on initial data and a nonlinear function.

Contribution

It investigates the existence and asymptotic behavior of solutions to a Kirchhoff total variation flow with the 1-Laplace operator, considering the influence of initial data and nonlinear function m.

Findings

Existence of solutions under certain conditions.

Asymptotic behavior near extinction time analyzed.

Dependence of solutions on initial data and m.

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.