Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912874834690048 |
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| author | Meyer, Felix Winkler, Michael |
| author_facet | Meyer, Felix Winkler, Michael |
| contents | The manuscript considers the model for conversion of mechanical energy into heat during acoustic wave propagation in the presence of temperature-dependent elastic parameters, as given by \[
\left\{ \begin{array}{l}
u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x, \\[1mm]
Θ_t = DΘ_{xx} + γ(Θ) u_{xt}^2.
\end{array} \right.
\qquad \qquad (\star) \] It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $Ω$, under the assumption that $γ\in C^2([0,\infty))$ is such that $γ>0$ and $γ'\ge 0$ on $[0,\infty)$ as well as \[
D\cdot (γ+D) \cdot γ'' + 2γγ'^2 \le 0
\qquad \mbox{on } [0,\infty), \] for all suitably regular initial data this problem admits a globally defined classical solution. This complements recent findings in the literature, according to which ($\star$) may admit solutions blowing up in finite time whenever $γ$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{dξ}{γ(ξ)} < \infty$.
Apart from that, it is found that if the additional assumption \[
a|Ω|^2 \le \frac{π^2 γ(0)}{1+\sqrt{1+\frac{γ(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting Meyer, Felix Winkler, Michael Analysis of PDEs 74F05, 35B40, 74A15, 80A17 The manuscript considers the model for conversion of mechanical energy into heat during acoustic wave propagation in the presence of temperature-dependent elastic parameters, as given by \[ \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x, \\[1mm] Θ_t = DΘ_{xx} + γ(Θ) u_{xt}^2. \end{array} \right. \qquad \qquad (\star) \] It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $Ω$, under the assumption that $γ\in C^2([0,\infty))$ is such that $γ>0$ and $γ'\ge 0$ on $[0,\infty)$ as well as \[ D\cdot (γ+D) \cdot γ'' + 2γγ'^2 \le 0 \qquad \mbox{on } [0,\infty), \] for all suitably regular initial data this problem admits a globally defined classical solution. This complements recent findings in the literature, according to which ($\star$) may admit solutions blowing up in finite time whenever $γ$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{dξ}{γ(ξ)} < \infty$. Apart from that, it is found that if the additional assumption \[ a|Ω|^2 \le \frac{π^2 γ(0)}{1+\sqrt{1+\frac{γ(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit. |
| title | Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting |
| topic | Analysis of PDEs 74F05, 35B40, 74A15, 80A17 |
| url | https://arxiv.org/abs/2602.04001 |