Characterizations and properties of solutions to parabolic problems of linear growth

Theo Elenius

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arXiv · 2505.16747

Characterizations and properties of solutions to parabolic problems of linear growth

Theo Elenius

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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.

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Theo Elenius. 2025-05-22. Characterizations and properties of solutions to parabolic problems of linear growth. https://arxiv.org/abs/2505.16747

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