The existence and dynamic properties of a parabolic nonlocal MEMS equation

Abstract

MSC:primary 35B40 secondary 35B05 35K50 35K20 Keywords:Nonlocal MEMS Pull-in voltage Parabolic nonlocal MEMS Asymptotic behaviour Quenching behaviour a b s t r a c tLet Ω ⊂ R n be a C 2 bounded domain and χ > 0 be a constant. We will prove the existence of constants λsuch that a solution exists for any 0 ≤ λ < λ * N and no solution exists for any λ > λ N where λ * is the pull-in voltage and w * is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ↗ λ * . Moreover λ N < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the solution of the parabolic nonlocal MEMS equationprove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.

Nonlocal Dynamic Problems with Singular Nonlinearities and Applications to MEMS

“…In this section we show that assumptions (i)-(v) of Theorem 4 are fulfilled for the penalized problem (13). As a consequence of [5] we have proved that for any…”

“…Here we consider a variant of the nonlocal contribution due to χ > 0, which generalizes the first-order approximation model (in Taylor's expansion, see [15]), in case of non-constant capacitance, corresponding to σ = 2. However, we will see that our approach allows more general nonlocal effects than the one considered here and in previous works [5,6,16,20,13,14]. We mention that evolution MEMS equations have been previously handled by different methods in [18,11,12], where existence results are obtained avoiding nonlocal contributions.…”

“…For parameters β, χ, γ = 0, the existence of the solution v ε to the stationary problem related to (13), follows from the existence of the solution v 0 corresponding to ε = 0, obtained in [4,2], by taking 0 < ε < (1 − ε 0 )/2 and setting v ε = v 0 + ε.…”

“…We mention that evolution MEMS equations have been previously handled by different methods in [18,11,12], where existence results are obtained avoiding nonlocal contributions. More recently, results for nonlocal parabolic problems are obtained in [13] whereas the second-order hyperbolic nonlocal MEMS equation is studied in the one-dimensional case in [14].…”

Global existence and quenching for a damped hyperbolic MEMS equation with the fringing field

“…Although we can not apply the comparison principle owing to the nonlocal term, in the elliptic and parabolic problem, there are similar results to those without nonlocal term, i.e., the global existence [14,16,20], its asymptotic behaviour [20], the quenching [14,16,20] and its stationary solution [14,16,20,22,33,36]. For a hyperbolic and damped hyperbolic operator A, we have the results of the global existence [15,22] and the quenching [22] for Ω = (0, 1).…”