$L^\infty$ blow-up in the Jordan-Moore-Gibson-Thompson equation
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909219560620032 |
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| author | Nikolić, Vanja Winkler, Michael |
| author_facet | Nikolić, Vanja Winkler, Michael |
| contents | The Jordan-Moore-Gibson-Thompson equation
\[
τu_{ttt} + αu_{tt} = βΔu_t + γΔu + (f(u))_{tt} \] is considered in a smoothly bounded domain $Ω\subset\mathbb{R}^n$ with $n\leq 3$, where $τ>0,β>0,γ>0$, and $α\in\mathbb{R}$.
Firstly, it is seen that under the assumption that $f\in C^3(\mathbb{R})$ is such that $f(0)=0$, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial-boundary value problem admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is such that \[
\mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(Ω)}=\infty. \] This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that \[
f''\ge 0
\mbox{ on $\mathbb{R}$,}
\qquad
\frac{f(ξ)}ξ \to +\infty
\mbox{ as $ξ\to +\infty$}
\qquad \mbox{and} \qquad
\int_{ξ_0}^\infty \frac{dξ}{f(ξ)} < \infty
\mbox{ for some $ξ_0>0$,} \] then for some initial data the above solution must undergo some finite-time $L^\infty$ blow-up in the style described above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^\infty$ blow-up in the Jordan-Moore-Gibson-Thompson equation Nikolić, Vanja Winkler, Michael Analysis of PDEs The Jordan-Moore-Gibson-Thompson equation \[ τu_{ttt} + αu_{tt} = βΔu_t + γΔu + (f(u))_{tt} \] is considered in a smoothly bounded domain $Ω\subset\mathbb{R}^n$ with $n\leq 3$, where $τ>0,β>0,γ>0$, and $α\in\mathbb{R}$. Firstly, it is seen that under the assumption that $f\in C^3(\mathbb{R})$ is such that $f(0)=0$, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial-boundary value problem admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is such that \[ \mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(Ω)}=\infty. \] This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that \[ f''\ge 0 \mbox{ on $\mathbb{R}$,} \qquad \frac{f(ξ)}ξ \to +\infty \mbox{ as $ξ\to +\infty$} \qquad \mbox{and} \qquad \int_{ξ_0}^\infty \frac{dξ}{f(ξ)} < \infty \mbox{ for some $ξ_0>0$,} \] then for some initial data the above solution must undergo some finite-time $L^\infty$ blow-up in the style described above. |
| title | $L^\infty$ blow-up in the Jordan-Moore-Gibson-Thompson equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.01595 |