Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909981375201280 |
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| author | Vitillaro, Enzo |
| author_facet | Vitillaro, Enzo |
| contents | The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-Δu+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\timesΩ$,}\\ u=0 &\text{on $(0,\infty)\times Γ_0$,}\\ u_{tt}+\partial_νu-Δ_Γu+Q(x,u_t)=g(x,u)\qquad &\text{on $(0,\infty)\times Γ_1$,}\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) &
\text{in $\overlineΩ$,} \end{cases}$$ where $Ω$ is a bounded open $C^1$ subset of $\mathbb{R}^N$, $N\ge 2$, $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a partition of $Γ$, $Γ_1\not=\emptyset$ being relatively open in $Γ$, $Δ_Γ$ denotes the Laplace--Beltrami operator on $Γ$, $ν$ is the outward normal to $Ω$, and the terms $P$ and $Q$ represent nonlinear damping terms, while $f$ and $g$ are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well--posedness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_08213 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources Vitillaro, Enzo Analysis of PDEs Mathematical Physics Functional Analysis 35L05, 35L10, 35L20, 35D30, 35Q74 The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-Δu+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\timesΩ$,}\\ u=0 &\text{on $(0,\infty)\times Γ_0$,}\\ u_{tt}+\partial_νu-Δ_Γu+Q(x,u_t)=g(x,u)\qquad &\text{on $(0,\infty)\times Γ_1$,}\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & \text{in $\overlineΩ$,} \end{cases}$$ where $Ω$ is a bounded open $C^1$ subset of $\mathbb{R}^N$, $N\ge 2$, $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a partition of $Γ$, $Γ_1\not=\emptyset$ being relatively open in $Γ$, $Δ_Γ$ denotes the Laplace--Beltrami operator on $Γ$, $ν$ is the outward normal to $Ω$, and the terms $P$ and $Q$ represent nonlinear damping terms, while $f$ and $g$ are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well--posedness. |
| title | Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis 35L05, 35L10, 35L20, 35D30, 35Q74 |
| url | https://arxiv.org/abs/2107.08213 |