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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> our definitions are equivalent with the definition based on the Anzellotti pairing.</p>

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Characterizations and properties of solutions to parabolic problems of linear growth

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月7日

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  • Theo Elenius

摘要

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 \(L^2\) L 2 our definitions are equivalent with the definition based on the Anzellotti pairing.