Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910921689923584 |
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| author | Winkler, Michael |
| author_facet | Winkler, Michael |
| contents | An initial-boundary value problem for \[
\left\{ \begin{array}{ll}
u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x,
\qquad & x\inΩ, \ t>0, \\[1mm]
Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt},
\qquad & x\inΩ, \ t>0,
\end{array} \right. \] is considered in an open bounded real interval $Ω$.
Under the assumption that $γ\in C^0([0,\infty))$ and $f\in C^0([0,\infty))$ are such that $f(0)=0$, and $k_γ\le γ\le K_γ$ as well as \[
|f(ξ)| \le K_f \cdot (ξ+1)^α
\qquad \mbox{for all } ξ\ge 0 \] with some $k_γ>0, K_γ>0, K_f>0$ and $α<\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities Winkler, Michael Analysis of PDEs 74H20, 74F05, 35L05, 35D30 An initial-boundary value problem for \[ \left\{ \begin{array}{ll} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \qquad & x\inΩ, \ t>0, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \qquad & x\inΩ, \ t>0, \end{array} \right. \] is considered in an open bounded real interval $Ω$. Under the assumption that $γ\in C^0([0,\infty))$ and $f\in C^0([0,\infty))$ are such that $f(0)=0$, and $k_γ\le γ\le K_γ$ as well as \[ |f(ξ)| \le K_f \cdot (ξ+1)^α \qquad \mbox{for all } ξ\ge 0 \] with some $k_γ>0, K_γ>0, K_f>0$ and $α<\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. |
| title | Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities |
| topic | Analysis of PDEs 74H20, 74F05, 35L05, 35D30 |
| url | https://arxiv.org/abs/2504.20480 |