Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid
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arXiv
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2026
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| _version_ | 1866911443664764928 |
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| author | Black, Tobias Winkler, Michael |
| author_facet | Black, Tobias Winkler, Michael |
| contents | This manuscript is concerned with the one-dimensional system \[
\begin{array}{l}
τu_{ttt} + αu_{tt} = b \big(γ(Θ) u_{xt}\big)_x + \big( γ(Θ) u_x\big)_x, \\[1mm]
Θ_t = D Θ_{xx} + bγ(Θ) u_{xt}^2,
\end{array} \] which is connected to the simplified modeling of heat generation in Zener type materials subject to stress from acoustic waves. Under the assumption that the coefficients $τ>0, b>0$ and $α\geq0$ satisfy \begin{align}\tag{$\star$}
αb >τ, \end{align} it is shown that for all $Θ_\star>0$ one can find $ν=ν(D,τ,α,b,Θ_\star,γ)>0$ such that an associated Neumann type initial-boundary value problem with Neumann data admits a unique time-global solution in a suitable framework of strong solvability whenever the initial temperature distribution fulfills $$\|Θ_0\|_{L^\infty(Ω)}\leq Θ_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_Ωu_{0xx}^2 + \int_Ω(u_{0t})_{xx}^2 + \int_Ω(u_{0tt})_x^2 < ν\quad\text{and}\quad
\|Θ_{0x}\|_{L^\infty(Ω)}
+ \|Θ_{0xx}\|_{L^\infty(Ω)}
< ν.$$ The constructed solution moreover features an exponential stabilization property for both components.
In particular, the parameter range described by ($\star$) coincides with the full stability regime known for the corresponding Moore--Gibson--Thompson equation despite the fairly strong nonlinear coupling to the temperature variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12171 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid Black, Tobias Winkler, Michael Analysis of PDEs 74H20, 74F05, 35L05, 35B40 This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} τu_{ttt} + αu_{tt} = b \big(γ(Θ) u_{xt}\big)_x + \big( γ(Θ) u_x\big)_x, \\[1mm] Θ_t = D Θ_{xx} + bγ(Θ) u_{xt}^2, \end{array} \] which is connected to the simplified modeling of heat generation in Zener type materials subject to stress from acoustic waves. Under the assumption that the coefficients $τ>0, b>0$ and $α\geq0$ satisfy \begin{align}\tag{$\star$} αb >τ, \end{align} it is shown that for all $Θ_\star>0$ one can find $ν=ν(D,τ,α,b,Θ_\star,γ)>0$ such that an associated Neumann type initial-boundary value problem with Neumann data admits a unique time-global solution in a suitable framework of strong solvability whenever the initial temperature distribution fulfills $$\|Θ_0\|_{L^\infty(Ω)}\leq Θ_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_Ωu_{0xx}^2 + \int_Ω(u_{0t})_{xx}^2 + \int_Ω(u_{0tt})_x^2 < ν\quad\text{and}\quad \|Θ_{0x}\|_{L^\infty(Ω)} + \|Θ_{0xx}\|_{L^\infty(Ω)} < ν.$$ The constructed solution moreover features an exponential stabilization property for both components. In particular, the parameter range described by ($\star$) coincides with the full stability regime known for the corresponding Moore--Gibson--Thompson equation despite the fairly strong nonlinear coupling to the temperature variable. |
| title | Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid |
| topic | Analysis of PDEs 74H20, 74F05, 35L05, 35B40 |
| url | https://arxiv.org/abs/2602.12171 |