The paper focuses on the lifespan of local-in-time solutions to the inhomogeneous semilinear heat equation driven by the fractional sub-Laplacian on the Heisenberg group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7781_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb{H}}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> : ∂tu+(−<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7781_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">${\Delta }_{{\mathbb{H}}^{N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>N</mi> </msup> </msub> </math></EquationSource> </InlineEquation>)su = |u|p+f in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7781_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb{H}}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> ×(0, T). In the subcritical range 1 < p < Q/(Q − 2s), we establish an upper bound on the maximal existence time, where Q = 2N + 2 is the homogeneous dimension of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7781_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb{H}}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>.

Lifespan of Local-In-Time Solutions to Semilinear Nonlocal Heat Equation in the Heisenberg Group

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发表日期：

2025年6月7日

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Priyank Oza

摘要

The paper focuses on the lifespan of local-in-time solutions to the inhomogeneous semilinear heat equation driven by the fractional sub-Laplacian on the Heisenberg group ${\mathbb{H}}^{N}$ $H^{N}$ : ∂_tu+(− ${\Delta }_{{\mathbb{H}}^{N}}$ $Δ_{H^{N}}$ )^su = |u|^p+f in ${\mathbb{H}}^{N}$ $H^{N}$ ×(0, T). In the subcritical range 1 < p < Q/(Q − 2s), we establish an upper bound on the maximal existence time, where Q = 2N + 2 is the homogeneous dimension of ${\mathbb{H}}^{N}$ $H^{N}$ .