The main goal of this work is to investigate the following quasilinear wave equation <Equation ID="Equ35"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3185_Article_Equ35.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="500" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} u_{tt}-div(|\nabla u|^{r(.)-2}\nabla u)-\Delta u_{t}+\alpha (t)|u_{t}|^{m(.)-2}u_{t}=0,\quad in\quad \Omega \times (0,T) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>-</mo> <msup> <mrow> <mi>d</mi> <mi>i</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> </mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mi>n</mi> <mspace width="1em" /> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3185_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded domain of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3185_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3185_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$T>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, m(.), r(.) are variable exponents and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3185_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the time-varying damping coefficient. We will establish several decay results under specific conditions on variable exponents and the time-varying coefficient. To illustrate our theoretical results, we give some numerical examples.

Decay results for a quasilinear wave equation with a nonlinear time-varying frictional damping of variable exponent

期刊论文
发表日期：

2025年4月3日

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Taklit Hamadouche,
Salim A. Messaoudi,
Mostafa Zahri

摘要

The main goal of this work is to investigate the following quasilinear wave equation $\begin{aligned} u_{tt}-div(|\nabla u|^{r(.)-2}\nabla u)-\Delta u_{t}+\alpha (t)|u_{t}|^{m(.)-2}u_{t}=0,\quad in\quad \Omega \times (0,T) \end{aligned}$ $\begin{matrix} u_{tt} - {d i v (| \nabla u |}^{r (.) - 2} \nabla u) - Δ u_{t} + α (t) {| u_{t} |}^{m (.) - 2} u_{t} = 0, i n Ω \times (0, T) \end{matrix}$ where $\Omega $ $Ω$ is a bounded domain of ${\mathbb {R}}^{n}$ $R^{n}$ , $T>0$ $T > 0$ , m(.), r(.) are variable exponents and $\alpha (t)$ $α (t)$ is the time-varying damping coefficient. We will establish several decay results under specific conditions on variable exponents and the time-varying coefficient. To illustrate our theoretical results, we give some numerical examples.