We prove existence of renormalized solution for a fractional (s, p)-Laplacian parabolic problem whose model is <Equation ID="Equ83"> <EquationSource Format="TEX">$(\mathcal {P})\left\{ \begin{aligned}&u_{t}+(-\varDelta )_{p}^{s}u(t,x)=\mu \text { in }Q:=(0,T)\times \varOmega , \\&u(0,x)=u_{0}(x)\text { in }\varOmega ,\ u(t,x)=0\text { on }\partial Q:=(0,T)\times \partial \varOmega , \end{aligned}\right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <msubsup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>Δ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> </mrow> <mi>s</mi> </msubsup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>μ</mi> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi>Q</mi> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mi>Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi>Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <mi>Q</mi> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mi>∂</mi> <mi>Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(-\varDelta )_{p}^{s}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>Δ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> </mrow> <mi>s</mi> </msubsup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> is the fractional (s, p)-Laplace operator (with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ps<N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi>s</mi> <mo><</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0<s<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>2-\frac{s}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> <mo>-</mo> <mfrac> <mi>s</mi> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>), <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_{0}\in L^{1}(\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu \in \mathcal {M}(Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (the vector space of all finite Radon measures in Q). We first prove some a priori estimates for the fractional parabolic (s, p)-capacity then we discuss the main properties of solutions without using the decomposition of the right-hand side.

On the notion of renormalized solution for the fractional (s, p)-Laplacian parabolic problem with diffuse measure data

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发表日期：

2026年2月24日

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Mohammed Abdellaoui,
Hicham Redwane

摘要

We prove existence of renormalized solution for a fractional (s, p)-Laplacian parabolic problem whose model is $(\mathcal {P})\left\{ \begin{aligned}&u_{t}+(-\varDelta )_{p}^{s}u(t,x)=\mu \text { in }Q:=(0,T)\times \varOmega , \\&u(0,x)=u_{0}(x)\text { in }\varOmega ,\ u(t,x)=0\text { on }\partial Q:=(0,T)\times \partial \varOmega , \end{aligned}\right. $ $(P) \{\begin{matrix} u_{t} + {(- Δ)}_{p}^{s} u (t, x) = μ in Q : = (0, T) \times Ω, \\ u (0, x) = u_{0} (x) in Ω, u (t, x) = 0 on \partial Q : = (0, T) \times \partial Ω, \end{matrix})$ where $(-\varDelta )_{p}^{s}u$ ${(- Δ)}_{p}^{s} u$ is the fractional (s, p)-Laplace operator (with $ps<N$ $p s < N$ , $0<s<1$ $0 < s < 1$ and $p>2-\frac{s}{N}$ $p > 2 - \frac{s}{N}$ ), $u_{0}\in L^{1}(\varOmega )$ $u_{0} \in L^{1} (Ω)$ and $\mu \in \mathcal {M}(Q)$ $μ \in M (Q)$ (the vector space of all finite Radon measures in Q). We first prove some a priori estimates for the fractional parabolic (s, p)-capacity then we discuss the main properties of solutions without using the decomposition of the right-hand side.