Characterizations and properties of solutions to parabolic problems of linear growth

TL;DR

This paper investigates weak solutions to a broad class of linear growth parabolic problems, establishing equivalences, approximation methods, and key properties like comparison principles and boundedness.

Contribution

It introduces a new framework for weak solutions of linear growth parabolic problems, including equivalence with variational solutions and novel approximation techniques.

Findings

Weak solutions are equivalent to variational solutions via stability.

Approximation of parabolic BV functions using mollification and Sobolev methods.

Established comparison principle and local boundedness for solutions.

Abstract

We consider notions of weak solutions to a general class of parabolic problems of linear growth, formulated independently of time regularity. Equivalence with variational solutions is established using a stability result for weak solutions. A key tool in our arguments is approximation of parabolic BV functions using time mollification and Sobolev approximations. We also prove a comparison principle and a local boundedness result for solutions. When the time derivative of the solution is in $L^2$ our definitions are equivalent with the definition based on the Anzellotti pairing.