Existence of solution for a class of heat equation involving the 1-Laplacian operator

TL;DR

This paper proves the existence of global solutions for a heat equation involving the 1-Laplacian operator by approximating with p-Laplacian problems and taking the limit as p approaches 1.

Contribution

It introduces an approximation method using p-Laplacian problems to establish global solutions for the 1-Laplacian heat equation.

Findings

Existence of global solutions for the heat equation with 1-Laplacian.

Use of p-Laplacian approximation technique.

Limit process as p approaches 1 to obtain results.

Abstract

This paper concerns the existence of global solutions for the following class of heat equation involving the 1-Laplacian operator of the Dirichlet problem $$ \left\{ \begin{array}{llc} u_{t}-\Delta_1 u=f(u) & \text{in}\ & \Omega\times (0, +\infty) , u =0 & \text{in} & \partial\Omega\times (0, +\infty), u(x,0)=u_{0}(x)& \text{in} &\Omega , \end{array}\right. \leqno{(P)} $$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N \geq 1$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some technical conditions, and $\Delta_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$ denotes the 1-Laplacian operator. The existence of global solution is done by using an approximation technique that consists in working with a class of $p$-Laplacian problem associated with $(P)$ and then taking the limit when $ p \to 1^+$ to get our results.