A Class of Initial-Boundary Value Problems Governed by Pseudo-Parabolic Weighted Total Variation Flows

TL;DR

This paper investigates a class of initial-boundary value problems driven by pseudo-parabolic total variation flows, analyzing the interplay between regularizing velocity effects and singular diffusion that may cause solution degeneration.

Contribution

It provides new mathematical results on the well-posedness and regularity of solutions for these complex pseudo-parabolic total variation flow problems.

Findings

Established conditions for solution existence and uniqueness.

Demonstrated the impact of singular diffusion on solution regularity.

Clarified the balance between regularization and degeneration effects.

Abstract

In this paper, we consider a class of initial-boundary value problems governed by pseudo-parabolic total variation flows. The principal characteristic of our problem lies in the velocity term of the diffusion flux, a feature that can bring about stronger regularity than what is found in standard parabolic PDEs. Meanwhile, our total variation flow contains singular diffusion, and this singularity may lead to a degeneration of the regularity of solution. The objective of this paper is to clarify the power balance between these conflicting effects. Consequently, we will present mathematical results concerning the well-posedness and regularity of the solution in the Main Theorems of this paper.