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<p>We establish the existence and uniqueness of a renormalized solution to an evolution equation featuring the non-local fractional <i>p</i>(<i>x</i>,&#xa0;<i>y</i>)-Laplacian and nonnegative <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_425_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-data. The definition of renormalized solutions is adapted to the non-local nature to bypass the use of chain rules which is unavailable. The fractional <i>p</i>(<i>x</i>,&#xa0;<i>y</i>)-Laplacian well encapsulates the fractional <i>p</i>-Laplacian with a constant exponent <i>p</i>. Hence our result extends [<CitationRef CitationID="CR25">25</CitationRef>] to the setting of variable exponents.</p>

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Renormalized solutions for a non-local evolution equation with variable exponent

  • 期刊论文

  • 发表日期：

    2025年5月22日

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  • Le Xuan Truong,
  • Nguyen Thanh Long,
  • Nguyen Ngoc Trong,
  • Tan Duc Do

摘要

We establish the existence and uniqueness of a renormalized solution to an evolution equation featuring the non-local fractional p(x, y)-Laplacian and nonnegative \(L^1\) L 1 -data. The definition of renormalized solutions is adapted to the non-local nature to bypass the use of chain rules which is unavailable. The fractional p(x, y)-Laplacian well encapsulates the fractional p-Laplacian with a constant exponent p. Hence our result extends [25] to the setting of variable exponents.