Well-posedness and asymptotic behavior of solutions to a second order nonlocal parabolic MEMS equation
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| Format: | Preprint |
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2026
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| author | Wei, Yufei Zhang, Yanyan |
| author_facet | Wei, Yufei Zhang, Yanyan |
| contents | We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-Δu=\fracλ{(1-u)^2\bigl(1+\int_Ω\frac{1}{1-u}\,dx\bigr)^2},\quad x\inΩ,\ t>0, \] where \(Ω\subset\mathbb{R}^N\) \((1\le N\le3)\) is a bounded smooth domain and \(λ>0\). Using operator semigroups and the contraction mapping principle, we prove local existence and give a quenching criterion. Under suitable smallness conditions on \(λ\) and the initial data, global existence and exponential convergence to the minimal steady state are obtained. Assuming the global solution stays uniformly away from the singularity \(u=1\), we show that the system forms a gradient system. By establishing analyticity of the energy and a Lojasiewicz--Simon inequality, we prove that the solution converges to a steady state with either exponential or algebraic rate depending on the Lojasiewicz exponent. Numerical experiments in 1D and 2D illustrate the results and support conjectures on the \(λ\)-dichotomy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07013 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness and asymptotic behavior of solutions to a second order nonlocal parabolic MEMS equation Wei, Yufei Zhang, Yanyan Analysis of PDEs 35B40, 35K55, 35A01, 35B44, 35R09 We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-Δu=\fracλ{(1-u)^2\bigl(1+\int_Ω\frac{1}{1-u}\,dx\bigr)^2},\quad x\inΩ,\ t>0, \] where \(Ω\subset\mathbb{R}^N\) \((1\le N\le3)\) is a bounded smooth domain and \(λ>0\). Using operator semigroups and the contraction mapping principle, we prove local existence and give a quenching criterion. Under suitable smallness conditions on \(λ\) and the initial data, global existence and exponential convergence to the minimal steady state are obtained. Assuming the global solution stays uniformly away from the singularity \(u=1\), we show that the system forms a gradient system. By establishing analyticity of the energy and a Lojasiewicz--Simon inequality, we prove that the solution converges to a steady state with either exponential or algebraic rate depending on the Lojasiewicz exponent. Numerical experiments in 1D and 2D illustrate the results and support conjectures on the \(λ\)-dichotomy. |
| title | Well-posedness and asymptotic behavior of solutions to a second order nonlocal parabolic MEMS equation |
| topic | Analysis of PDEs 35B40, 35K55, 35A01, 35B44, 35R09 |
| url | https://arxiv.org/abs/2603.07013 |