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<p>This paper aims to investigate the initial boundary value problem (<InternalRef RefID="Equ1">1.1</InternalRef>) below, where the stochastic quasilinear viscoelastic wave equation involves a nonlinear damping of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_738_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(|u_{t}|^{m-2}u_{t}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_738_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">\(u|u|^{q-2}ln |u|^{q}\)</EquationSource> </InlineEquation> as a source term. The equation being driven by an additive noise. Using an appropriate energy inequality, we prove that finite time blow-up is possible with positive probability for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_738_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(q&gt;m\)</EquationSource> </InlineEquation> when the initial energy is sufficiently negative. In addition, we prove that for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_738_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\ge m\)</EquationSource> </InlineEquation>, the local solution can be extended for all time.</p>

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Blow-Up and Energy Decay for the Solution of a Stochastic Viscoelastic Wave Equation with Memory Owning a Nonlinear Damping and a Logarithmic Source Terms

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Fatna Bensaber,
  • Amina Benramdane,
  • Nadia Mezouar,
  • Salah Boulaaras,
  • Ibrahim Mekawy

摘要

This paper aims to investigate the initial boundary value problem (1.1) below, where the stochastic quasilinear viscoelastic wave equation involves a nonlinear damping of the form \(|u_{t}|^{m-2}u_{t}\) and \(u|u|^{q-2}ln |u|^{q}\) as a source term. The equation being driven by an additive noise. Using an appropriate energy inequality, we prove that finite time blow-up is possible with positive probability for \(q>m\) when the initial energy is sufficiently negative. In addition, we prove that for \(q\ge m\) , the local solution can be extended for all time.