<p>This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. The model equation is <Equation ID="Equ195"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1628_Article_Equ195.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="434" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \partial _t u-\operatorname {div}(u^{m-1}|Du|^{p-2}Du)=f\qquad \text {in}\quad \Omega \times (-T,0)\subset {\mathbb {R}}^{n+1}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mrow> <mi>u</mi> <mo>-</mo> <mo>div</mo> <mo stretchy="false">(</mo> </mrow> <msup> <mi>u</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mspace width="2em" /> <mtext>in</mtext> <mspace width="1em" /> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>T</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> </mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Here, we consider the case <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1628_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1628_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$2<p<n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. We establish a continuity estimate for <i>u</i> in terms of elliptic Riesz potentials on the right-hand side of the equation.</p>

Continuity estimates for doubly degenerate parabolic equations with lower-order terms via nonlinear potentials

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发表日期：

2025年3月22日

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Qifan Li

摘要

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. The model equation is $\begin{aligned} \partial _t u-\operatorname {div}(u^{m-1}|Du|^{p-2}Du)=f\qquad \text {in}\quad \Omega \times (-T,0)\subset {\mathbb {R}}^{n+1}. \end{aligned}$ $\begin{matrix} \partial_{t} u - div (u^{m - 1} {| D u |}^{p - 2} D u) = f in Ω \times (- T, 0) \subset R^{n + 1} . \end{matrix}$ Here, we consider the case $m>1$ $m > 1$ and $2<p<n$ $2 < p < n$ . We establish a continuity estimate for u in terms of elliptic Riesz potentials on the right-hand side of the equation.