<p>This manuscript is concerned with the evolution system <Equation ID="Equ167"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{l}u_{ttt} + \alpha u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + \big ( \widehat{\gamma }(\Theta ) u_x\big )_x, \\ \Theta _t = D \Theta _{xx} + \Gamma (\Theta ) u_{xt}^2, \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">ttt</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>γ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xt</mi> </mrow> </msub> <msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>x</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi mathvariant="normal">Θ</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>D</mi> <msub> <mi mathvariant="normal">Θ</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>u</mi> <mrow> <mi mathvariant="italic">xt</mi> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type.</p><p>Under the assumptions that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma , \widehat{\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>,</mo> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> are sufficiently smooth with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma>0, \widehat{\gamma }>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">^</mo> </mover> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.</p>

Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

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2026年6月25日

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Leander Claes,
Michael Winkler

摘要

This manuscript is concerned with the evolution system $\begin{aligned} \left\{ \begin{array}{l}u_{ttt} + \alpha u_{tt} = \big (\gamma (\Theta ) u_{xt}\big )_x + \big ( \widehat{\gamma }(\Theta ) u_x\big )_x, \\ \Theta _t = D \Theta _{xx} + \Gamma (\Theta ) u_{xt}^2, \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{ttt} + α u_{tt} = (γ (Θ) u_{xt})_{x} + (\hat{γ} (Θ) u_{x})_{x}, \\ Θ_{t} = D Θ_{xx} + Γ (Θ) u_{xt}^{2}, \end{matrix}) \end{matrix}$ which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type.

Under the assumptions that $D>0$ $D > 0$ and $\alpha \ge 0$ $α \geq 0$ , and that $\gamma , \widehat{\gamma }$ $γ, \hat{γ}$ and $\Gamma $ $Γ$ are sufficiently smooth with $\gamma>0, \widehat{\gamma }>0$ $γ > 0, \hat{γ} > 0$ and $\Gamma \ge 0$ $Γ \geq 0$ on $[0,\infty )$ $[0, \infty)$ , for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.