In the article we study the solution u(x, t) of the Cauchy problem of linear damped fractional wave equation. We prove that u(x, t) has some sharp boundedness estimates on the Triebel–Lizorkin space. The proof of the necessity part is based on obtaining the precise asymptotic forms of the kernels of operators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3294_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">${\rm e}^{-t} \cosh(t{\sqrt L})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mi>cosh</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mrow> <msqrt> <mi>L</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3294_Article_IEq2.gif" Format="GIF" Height="30" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">${\rm e}^{-t} {{\sinh(t{\sqrt L})} \over {\sqrt L}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mrow> <mfrac> <mrow> <mi>sinh</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mrow> <msqrt> <mi>L</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <msqrt> <mi>L</mi> </msqrt> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> with L = 1 − ∣Δ∣α, where Δ is the Laplacian, as well as the method of stationary phase. Additionally, we study the Riesz mean of the solution and show its convergence in the Triebel–Lizorkin space norm.

Solution of Linear Damped Fractional Wave Equation on Triebel–Lizorkin Spaces

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发表日期：

2025年7月15日

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Meizhong Wang,
Jiecheng Chen,
Dashan Fan,
Ziyao Liu

摘要

In the article we study the solution u(x, t) of the Cauchy problem of linear damped fractional wave equation. We prove that u(x, t) has some sharp boundedness estimates on the Triebel–Lizorkin space. The proof of the necessity part is based on obtaining the precise asymptotic forms of the kernels of operators ${\rm e}^{-t} \cosh(t{\sqrt L})$ $e^{- t} \cosh (t \sqrt{L})$ and ${\rm e}^{-t} {{\sinh(t{\sqrt L})} \over {\sqrt L}}$ $e^{- t} \frac{\sinh (t \sqrt{L})}{\sqrt{L}}$ with L = 1 − ∣Δ∣^α, where Δ is the Laplacian, as well as the method of stationary phase. Additionally, we study the Riesz mean of the solution and show its convergence in the Triebel–Lizorkin space norm.