On A Parabolic Equation in MEMS with An External PressureThe parabolic problem $u_t-Δu=\frac{λf(x)}{(1-u)^2}+P$ on a
bounded domain $Ω$ of $R^n$ with Dirichlet boundary condition models the
microelectromechanical systems(MEMS) device with an external pressure term. In
this paper, we classify the behavior of the solution to this equation. We first
show that under certain initial conditions, there exists critical constants
$P^*$ and $λ_P^*$ such that when $0\leq P\leq P^*$, $0<λ\leq
λ_P^*$, there exists a global solution, while for $0\leq P\leq
P^*,λ>λ_P^*$ or $P>P^*$, the solution quenches in finite time. The
estimate of voltage $λ_P^*$, quenching time $T$ and pressure term $P^*$
are investigated. The quenching set $Σ$ is proved to be a compact subset
of $Ω$ with an additional condition, provided $Ω\subset R^n$ is a
convex bounded set. In particular, if $Ω$ is radially symmetric, then the
origin is the only quenching point. Furthermore, we not only derive the
two-side bound estimate for the quenching solution, but also study the
asymptotic behavior of the quenching solution in finite time.
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