Global and touchdown behaviour of the generalized MEMS device equation
| dc.creator | Hui, Kin Ming | |
| dc.date | 2008-08-01 | |
| dc.date.accessioned | 2026-07-07T09:54:16Z | |
| dc.date.available | 2026-07-07T09:54:16Z | |
| dc.description | We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation $u_t =Δu+λf(x)/g(u)$, $u<1$, in $Ω\times (0,\infty)$, $u(x,t)=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, where $Ω\subset\Bbb{R}^n$ is a bounded domain, $λ>0$ is a constant, $0\le f\in C^α(\overlineΩ)$, $f\not\equiv 0$, for some constant $0<α<1$, $0<g\in C^2((-\infty,1))$ such that $g'(s)\le 0$ for any $s<1$ and $u_0\in L^1(Ω)$ with $u_0\le a<1$ for some constant $a$. We prove that there exists a constant $λ^{\ast}=λ^{\ast}(Ω, f,g)>0$ such that the associated stationary problem has a solution for any $0\leλ<λ^*$ and has no solution for any $λ>λ^*$. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as $t\to\infty$ for any $0\leλ<λ^*$. We also obtain various conditions for the existence of a touchdown time $T>0$ for the solution $u$. That is a time $T>0$ such that $\lim_{t\nearrow T}\sup_Ωu(\cdot,t)=1$. | |
| dc.description | 25 pages | |
| dc.identifier | https://arxiv.org/abs/0808.0110 | |
| dc.identifier | http://arxiv.org/abs/0808.0110 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/166248 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35B40, 35B05, 35K50, 35K20 | |
| dc.title | Global and touchdown behaviour of the generalized MEMS device equation | |
| dc.type | text |