<p>We extend the concept of borderline solution introduced by Ni et al. (J Differ Equ 54:97–120, 1984) to a semilinear parabolic equation of the form <Equation ID="Equ49"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3811_Article_Equ49.gif" Format="GIF" Height="39" Rendition="HTML" Resolution="72" Type="Linedraw" Width="495" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} u_t-\Sigma _{i,j=1}^na^{ij}(x,t)\frac{\partial ^2u}{\partial x^i\partial x^j}+\Sigma _{i=1}^nb^i(x,t)\frac{\partial u}{\partial x^i}+C(x,t)u=K(x,t)f(u). \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>-</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>a</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <msup> <mi>x</mi> <mi>i</mi> </msup> <mi>∂</mi> <msup> <mi>x</mi> <mi>j</mi> </msup> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>b</mi> <mi>i</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <msup> <mi>x</mi> <mi>i</mi> </msup> </mrow> </mfrac> <mo>+</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3811_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$p>1+\frac{4}{n}, n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>4</mn> <mi>n</mi> </mfrac> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the solution is regular outside a compact set of Hausdorff dimension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3811_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$n-\frac{4}{p-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mfrac> <mn>4</mn> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

Partial regularity results for borderline solution to a semilinear parabolic equation

期刊论文
发表日期：

2025年7月11日

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Shi-Zhong Du,
Xu-Qian Fan

摘要

We extend the concept of borderline solution introduced by Ni et al. (J Differ Equ 54:97–120, 1984) to a semilinear parabolic equation of the form $\begin{aligned} u_t-\Sigma _{i,j=1}^na^{ij}(x,t)\frac{\partial ^2u}{\partial x^i\partial x^j}+\Sigma _{i=1}^nb^i(x,t)\frac{\partial u}{\partial x^i}+C(x,t)u=K(x,t)f(u). \end{aligned}$ $\begin{matrix} u_{t} - Σ_{i, j = 1}^{n} a^{ij} (x, t) \frac{\partial^{2} u}{\partial x^{i} \partial x^{j}} + Σ_{i = 1}^{n} b^{i} (x, t) \frac{\partial u}{\partial x^{i}} + C (x, t) u = K (x, t) f (u) . \end{matrix}$ For $p>1+\frac{4}{n}, n\ge 1$ $p > 1 + \frac{4}{n}, n \geq 1$ , we show that the solution is regular outside a compact set of Hausdorff dimension $n-\frac{4}{p-1}$ $n - \frac{4}{p - 1}$ .