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<p>Inspired by Zhang (Sci China Math 64:1201–1230, 2021, J Math Anal Appl 485(1):123770, 2020), we derive a series of local gradient estimates for positive solutions of a weighted parabolic partial differential equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1788_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta _{\phi } u = u_t\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> </msub> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> under the weighted Yamabe flow. As a consequence, we derive some global gradient estimates, Harnack inequalities, two-side Gaussian bounds for heat kernel, and the monotonicity of parabolic frequency.</p>

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Gradient estimates for a weighted heat equation under weighted Yamabe flow and applications

  • 期刊论文

  • 发表日期：

    2025年9月13日

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  • Nguyen Tien Manh

摘要

Inspired by Zhang (Sci China Math 64:1201–1230, 2021, J Math Anal Appl 485(1):123770, 2020), we derive a series of local gradient estimates for positive solutions of a weighted parabolic partial differential equation \(\Delta _{\phi } u = u_t\) Δ ϕ u = u t under the weighted Yamabe flow. As a consequence, we derive some global gradient estimates, Harnack inequalities, two-side Gaussian bounds for heat kernel, and the monotonicity of parabolic frequency.