<p>This study focuses on the existence, uniqueness, and quenching behavior of solution to the time-space fractional Kawarada problem, where the time derivative is the Caputo-Hadamard derivative and the spatial derivative is the fractional Laplacian. The mild solution represented by Fox <i>H</i>-function, based on the fundamental solution, is considered in space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_384_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </InlineMediaObject> <EquationSource Format="TEX">$C\left( [a, T], L^r(\mathbb {R}^d)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <msup> <mi>L</mi> <mi>r</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. We use the fractional maximum principles to prove <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_384_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$u(\textrm{x},t)\ge u_a(\textrm{x})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mtext>x</mtext> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msub> <mi>u</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mtext>x</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for the positive initial value. Then the relationship between quenching phenomena and the size of domain is examined. Finally, the finite difference scheme is established for solving the quenching solution to the considered problem in one and two space dimensions. The numerical simulations show the effectiveness and feasibility of the theoretical analysis.</p>

Analysis and computation for quenching solution to the time-space fractional Kawarada problem

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发表日期：

2025年3月14日

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Dingding Cao,
Changpin Li

摘要

This study focuses on the existence, uniqueness, and quenching behavior of solution to the time-space fractional Kawarada problem, where the time derivative is the Caputo-Hadamard derivative and the spatial derivative is the fractional Laplacian. The mild solution represented by Fox H-function, based on the fundamental solution, is considered in space $C\left( [a, T], L^r(\mathbb {R}^d)\right) $ $C ([a, T], L^{r} (R^{d}))$ . We use the fractional maximum principles to prove $u(\textrm{x},t)\ge u_a(\textrm{x})$ $u (x, t) \geq u_{a} (x)$ for the positive initial value. Then the relationship between quenching phenomena and the size of domain is examined. Finally, the finite difference scheme is established for solving the quenching solution to the considered problem in one and two space dimensions. The numerical simulations show the effectiveness and feasibility of the theoretical analysis.