ON SOME TOUCHDOWN BEHAVIORS OF THE GENERALIZED MEMS DEVICE EQUATION

. We study the quenching behaviors for the generalized microelec-tromechanical system (MEMS) equation u t − ∆ u = λρ ( x ) f ( u ), 0 < u < A ( A = 1 or + ∞ ), in Ω × (0 , + ∞ ), u ( x,t ) = 0 on ∂ Ω × (0 , + ∞ ), u ( x, 0) = u 0 ( x ) ∈ [0 ,A ) in Ω, where λ > 0, Ω ⊂ R N is a bounded domain, 0 ≤ ρ ( x ) ∈ C α (Ω), ρ (cid:54)≡ 0, for some constant 0 < α < 1, 0 < f ∈ C 2 ((0 ,A )) such that f (cid:48) ( s ) ≥ 0, f (cid:48)(cid:48) ( s ) ≥ 0 for any s ∈ [0 ,A ) and u 0 is smooth, u 0 = 0 on ∂ Ω. It is well known that quenching does occur and corresponds to a touchdown phenomenon. We es-tablish an interesting quenching rate, and based on which we then prove that touchdown cannot occur at zero points of ρ ( x ) or at the boundary of Ω, without the assumption of compactness of the touchdown set.


Introduction.
Let Ω be a bounded domain of class C 2+ν for some ν ∈ (0, 1) in R N . Let 0 ≤ ρ ∈ C α (Ω) for some constant 0 < α < 1 and ρ ≡ 0 in Ω (1) and let f > 0 satisfy one of the following two conditions: In this paper we consider the generalized MEMS equation where λ > 0 and the initial data u 0 (x) is smooth satisfying ∆u 0 + ρ(x)f (u 0 ) ≥ 0. When f (u) = (1 − u) −2 and u 0 ≡ 0, (3) reduces to the evolution Micro-electromechanical systems (MEMS) equations which were studied extensively in [2,3,5,6,10,11,14]. For the details of background and derivation of MEMS model, one 2448 QI WANG can refer to [16]. The equation (3) with f (u) = (1 − u) −p (p > 0) was studied in [9,13,17]. We know by the standard parabolic theory or by [12] that there exists a unique classical solution u of (3) in a short time interval. Also, by the strong maximum principle, we can see u > 0 in Ω for t > 0. Moreover, u can be continued as long as sup x∈Ω u(x, t) < A. It is well known (see, e.g., [1,18] and the references therein) that for any given ρ, there exists a critical value λ * > 0 such that if λ ∈ (0, λ * ), the solution to (3) is global with u 0 = 0; while for λ > λ * , the solution to (3) will reach the value A at finite time T , i.e. the so called quenching or touchdown phenomenon occurs. The more precise definition of the quenching time T is The corresponding quenching set is defined as A point x = x 0 ∈ Σ is called a quenching point. It is essential to understand the quenching phenomenon, such as the quenching set Σ, the quenching points, the rate of the quenching solution. Some interesting results have been obtained in several recent works (see for example [6,7,10] and the references therein). Note that a long-standing open problem is to decide how to describe the quenching points or quenching set. In [6,10], under the assumption that the quenching set is a compact subset of Ω, it is shown that x 0 is not a quenching point if ρ(x 0 ) = 0. On the other hand, the compactness assumption was proved in [10] by adapting a moving plane argument from [4,8] when ρ is constant, or more generally, when ρ is nonincreasing as one approaches the boundary. In this paper, inspired by [9], we study the problem (3) and give some results about the quenching points and the quenching set, as well as for the case of general ρ, in any space dimension. For simplicity, we denote d(x) = dist(x, ∂Ω), x ∈ Ω, the function distance to the boundary. Our main conclusions is the following. Theorem 1.1. Assume (1), (R)(or (S)) and let the solution u of problem (3) be such that T < +∞. Then there exist c 1 > 0, ε > 0 (independent of x, t) and the function h(z), C(y) such that ≥ C(y) > 0, for any y ∈ (0, +∞), Theorem 1.2. Assume (1), (R)(or (S)) and let the solution u of problem (3) be such that T < +∞. If x 0 ∈ Ω is such that ρ(x 0 ) = 0, then x 0 is not a quenching point.
As a consequence of Theorem 1.1 and of suitable comparison arguments, we are able to obtain two further criteria for the quenching set to be compact. We note that we do not require any convexity of the domain.
where h is defined as in Theorem 1.1. Then quenching does not occur near the boundary, i.e., Σ ⊂ Ω.
As a consequence of our results, we see that the latter has to be part of the positive set of the function ρ. In fact it would be desirable to gain further information about the structure of the quenching set, but this seems a difficult mathematical problem for nonconstant ρ, even in one space dimension.
2. Proof of Theorem 1.1. Theorem 1.1 will be proved via a nontrivial modification of the method in [4,8]. Consider the function where ε > 0 and a(x) ∈ C 2 (Ω) is an auxiliary function such that a ≥ 0, a| ∂Ω = 0, hence J = 0 on ∂Ω × (0, T ). The function g(u) ≥ 0 will be decided later. Setting It follows from the Hopf lemma and the strong maximum principle that for any Then the comparison principle gives that For any 0 < t 0 < T , we next claim that there exist ε > 0, suitable a(x) and g, such that Indeed, direct calculations imply that where the Cauchy-Schwarz inequality is applied.
Here, θ > 0, b ∈ (0, b 0 ), and γ > 0 is a positive constant such that where ε, c 1 , h are given in Theorem 1.1. Let v(x) be such that ρ(x)) = 0, then we can deduce that by taking sufficiently small b and θ.

Notice that
It then follows from (27) that w(x, 0) ≥ u 0 (x) for all x ∈ B(x 0 , b). On the other hand since (27) gives w| ∂B(x0,b)×(0,T ) ≥ u| ∂B(x0,b)×(0,T ) . Hence the comparison principle yields that u ≤ w in B(x 0 , b) × (0, T ). Note also that min φ > 0. This shows that which implies that sup u < A and hence x = x 0 is not a quenching point. So we are done. Since Ω is of class C 2+ν , ∂Ω satisfies exterior ball condition. Take any point x 0 ∈ ∂Ω, and without loss of generality assume that x 0 = 0 with B(0, |x 0 |) Ω = ∅. We next look for a suitable supersolution. Consider the function z(x) := K(d − r), r = |x|, with d > |x 0 | to be chosen, and 0 < r < d.
It is easy to see that z r = −K (d − r), z rr = K (d − r). Define now

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Then we have in Θ that Because (34) gives that K ≤ 0, K ≤ 0, we compute K f (K) < 0, and hence we obtain from (36) that Letting d > |x 0 | close to x 0 then yields that −∆z−λρ(x)f (z) ≥ 0 in Θ. Similarly, by taking d possibly closer to |x 0 |, we get that z( in Θ, and also for x ∈ ∂Ω ∂Θ, z(x) ≥ K(d − |x 0 |) ≥ 0. On the other hand, z(x) = K(0) = 1 > u(x, t) for x ∈ {|x| = d} ∂Θ and t ∈ (0, T ). Therefore it follows from the comparison principle that z ≥ u on Θ × (0, T ) and hence x 0 is not a quenching point. So the theorem follows. Define Ω η := {x ∈ Ω : d(x) < η}. Then there exists η 0 > 0 such that Ω η is a smooth bounded domain for all η ∈ (0, η 0 ), due to Ω being a smooth domain, and we have Consider the function w(x, t) as follows, Here the function m satisfies and φ(x) is to be decided later. Let ψ η be the unique solution of the problem Then the strong maximum principle and the standard elliptic regularity theory imply that ψ η is smooth and 0 < ψ η < 1 in Ω η . We introduce now an auxiliary function R(y) such that and set φ(x) = κR(ψ η (x)), where κ > 0 and η ∈ (0, η 0 ) are constants to be chosen later. Then one can see Consequently, 0 ≤ φ ≤ κR(1).