Some qualitative properties for the Kirchhoff total variation flow

Abstract

The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem.

Keywords:

• Kirchhoff total variation flow
• extinction time

2000 Mathematics Subject Classifications:

• 35K55
• 35B40

Acknowledgments

The author would like to thank Professor Claudianor Alves for several interesting and useful discussions on this topic. The author also would like to thank the reviewers for their time to read the manuscript and kind comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.