In this paper, we investigate the existence of weak solutions for the p(x)-parabolic equation with logarithmic nonlinearity of the form <Equation ID="Equ82"> <EquationSource Format="TEX">$\begin{aligned} v_t-A(v)=\vert v\vert ^{q(x)-2}v\log (\vert v \vert )+\vert v\vert ^{\alpha (x)-2}v, \; \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>-</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>v</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mo>log</mo> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>v</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>v</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mo>,</mo> <mspace width="0.277778em" /> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where A is an elliptic operator that maps <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W_{0}^{1,p(\cdot )}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into its dual space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$W^{-1,p^{\prime }(\cdot )}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also obtain an upper bound for blow-up time of weak solutions under some suitable conditions. The study of such problem will be in the setting of Lebesgue-Sobolev spaces with variable exponents. Our proof is based on Galerkin approximation method and concavity method.

Existence and blow-up phenomena for a diffusion equation with variable exponent and logarithmic nonlinearity

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发表日期：

2026年2月3日

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Abdellatif Bouhal,
Youssef El Hadfi,
Hichem Khelifi

摘要

In this paper, we investigate the existence of weak solutions for the p(x)-parabolic equation with logarithmic nonlinearity of the form $\begin{aligned} v_t-A(v)=\vert v\vert ^{q(x)-2}v\log (\vert v \vert )+\vert v\vert ^{\alpha (x)-2}v, \; \end{aligned}$ $\begin{matrix} v_{t} - A (v) = {| v |}^{q (x) - 2} v log (| v |) + {| v |}^{α (x) - 2} v, \end{matrix}$ where A is an elliptic operator that maps $W_{0}^{1,p(\cdot )}(\Omega )$ $W_{0}^{1, p (\cdot)} (Ω)$ into its dual space $W^{-1,p^{\prime }(\cdot )}(\Omega )$ $W^{- 1, p^{'} (\cdot)} (Ω)$ . We also obtain an upper bound for blow-up time of weak solutions under some suitable conditions. The study of such problem will be in the setting of Lebesgue-Sobolev spaces with variable exponents. Our proof is based on Galerkin approximation method and concavity method.