Nonradial Quenching Profile for a MEMS Model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915700553023488 |
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| author | Liao, Hsuan-Lin Nguyen, Van Tien |
| author_facet | Liao, Hsuan-Lin Nguyen, Van Tien |
| contents | We construct a quenching solution to the parabolic MEMS model \[ u_t = Δu - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and $T > 0$ denotes the quenching time. The constructed solution quenches only at the origin and admits the final profile \[ u(x,T) \sim \left(x_1^2 x_2^2 + θ(x_1^6 + x_2^6)\right)^{\frac{1}{3}} \quad \text{as } |x| \to 0, \] where $θ\in (0, θ^*)$ for some $θ^* > 0$. To our knowledge, this is the first example of a quenching solution with a genuinely non-radial profile. The proof relies on the construction of a good approximate solution, using a perturbative expansion in self-similar variables. We then justify the true solution that remains close to this approximation through a spectral analysis combined with a robust energy method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonradial Quenching Profile for a MEMS Model Liao, Hsuan-Lin Nguyen, Van Tien Analysis of PDEs 35B44, 35K55, 35B40 We construct a quenching solution to the parabolic MEMS model \[ u_t = Δu - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and $T > 0$ denotes the quenching time. The constructed solution quenches only at the origin and admits the final profile \[ u(x,T) \sim \left(x_1^2 x_2^2 + θ(x_1^6 + x_2^6)\right)^{\frac{1}{3}} \quad \text{as } |x| \to 0, \] where $θ\in (0, θ^*)$ for some $θ^* > 0$. To our knowledge, this is the first example of a quenching solution with a genuinely non-radial profile. The proof relies on the construction of a good approximate solution, using a perturbative expansion in self-similar variables. We then justify the true solution that remains close to this approximation through a spectral analysis combined with a robust energy method. |
| title | Nonradial Quenching Profile for a MEMS Model |
| topic | Analysis of PDEs 35B44, 35K55, 35B40 |
| url | https://arxiv.org/abs/2510.15246 |