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<p>This work addresses a value problem concerning a system of high-order nonlinear equations with viscoelastic terms acting in both equations and homogeneous Dirichlet conditions. Initially, we demonstrate that the system has a weak local solution by combining fixed point theory and Galerkin’s method. Under certain conditions on the variable exponents <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3250_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$p(.)$</EquationSource> </InlineEquation>, as well as conditions related to the kernel functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3250_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>η</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>≤</mo> <mi>μ</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></EquationSource> <EquationSource Format="TEX">$\eta \left ( t\right ) \leq \mu \left ( t\right ) $</EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3250_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$t\geq 0$</EquationSource> </InlineEquation>, we establish that the solution with positive initial energy exhibits exponential growth. In addition, we study some numerical examples to illustrate our theoretical results and demonstrate the effectiveness of our new approach.</p>

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Exponential growth of positive initial energy for a system of higher-order viscoelastic wave equations with variable exponents

  • 期刊论文

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  • 发表日期：

    2025年1月8日

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  • Amar Ouaoua,
  • Wissem Boughamsa,
  • Salah Boulaaras

摘要

This work addresses a value problem concerning a system of high-order nonlinear equations with viscoelastic terms acting in both equations and homogeneous Dirichlet conditions. Initially, we demonstrate that the system has a weak local solution by combining fixed point theory and Galerkin’s method. Under certain conditions on the variable exponents p ( . ) $p(.)$ , as well as conditions related to the kernel functions η ( t ) ≤ μ ( t ) $\eta \left ( t\right ) \leq \mu \left ( t\right ) $ , for all t ≥ 0 $t\geq 0$ , we establish that the solution with positive initial energy exhibits exponential growth. In addition, we study some numerical examples to illustrate our theoretical results and demonstrate the effectiveness of our new approach.