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<p>We are concerned with the initial-boundary-value problem for a coupled system of viscoelastic equations with variable exponents. Under assumptions on initial data and log-Hölder continuous exponents <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(m(x), r(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove local existence of weak solutions to the initial-boundary-value problem using the fixed point theory, the Galerkin method, the auxiliary system approach, and other techniques. Moreover, we estimate the upper bound of the lifespan of the blow-up solutions to the coupled system using energy methods, differential inequalities, and new estimation techniques.</p>

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Existence and blow-up in a coupled system of viscoelastic equations with variable exponents

  • 期刊论文

  • 发表日期：

    2026年7月14日

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  • Jin Liang,
  • Zhi-Cheng Xiong

摘要

We are concerned with the initial-boundary-value problem for a coupled system of viscoelastic equations with variable exponents. Under assumptions on initial data and log-Hölder continuous exponents \(m(x), r(x)\) m ( x ) , r ( x ) , we prove local existence of weak solutions to the initial-boundary-value problem using the fixed point theory, the Galerkin method, the auxiliary system approach, and other techniques. Moreover, we estimate the upper bound of the lifespan of the blow-up solutions to the coupled system using energy methods, differential inequalities, and new estimation techniques.