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<p>We establish a class of pointwise estimates for weak solutions to mixed local and nonlocal parabolic equations involving measure data and merely measurable coefficients via caloric Riesz potentials. Such estimates effectively bound the sizes and oscillations of weak solutions, respectively. The proof relies on demonstrating a new local Hölder estimate with an optimal <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2008_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-Tail for weak solutions to the corresponding homogeneous problem, which remarkably extends the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2008_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-Tail in previous work. It is worth mentioning that our main results capture both local and nonlocal features of the double phase parabolic equations and, more importantly, remain valid for SOLA (Solutions Obtained by Limit of Approximations).</p>

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Pointwise and Oscillation Estimates Via Riesz Potentials for Mixed Local and Nonlocal Parabolic Equations

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Lingwei Ma,
  • Qi Xiong,
  • Zhenqiu Zhang

摘要

We establish a class of pointwise estimates for weak solutions to mixed local and nonlocal parabolic equations involving measure data and merely measurable coefficients via caloric Riesz potentials. Such estimates effectively bound the sizes and oscillations of weak solutions, respectively. The proof relies on demonstrating a new local Hölder estimate with an optimal \(L^q\) L q -Tail for weak solutions to the corresponding homogeneous problem, which remarkably extends the \(L^\infty \) L ∞ -Tail in previous work. It is worth mentioning that our main results capture both local and nonlocal features of the double phase parabolic equations and, more importantly, remain valid for SOLA (Solutions Obtained by Limit of Approximations).