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In this paper, we consider a class of initial-boundary value problemsgoverned by pseudo-parabolic total variation flows. The principalcharacteristic of our problem lies in the velocity term of the diffusion flux,a feature that can bring about stronger regularity than what is found instandard parabolic PDEs. Meanwhile, our total variation flow contains singulardiffusion, and this singularity may lead to a degeneration of the regularity ofsolution. The objective of this paper is to clarify the power balance betweenthese conflicting effects. Consequently, we will present mathematical resultsconcerning the well-posedness and regularity of the solution in the MainTheorems of this paper.

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