Touchdown and related problems in electrostatic MEMS device equation

Nikos I. Kavallaris 0 1 2 Tosiya Miyasita 0 1 2 Takashi Suzuki 0 1 2 0 Takashi Suzuki Division of Mathematical Science Graduate School of Engineering Science Osaka University Toyonaka 560-8531 Japan 1 Tosiya Miyasita Department of Mathematics Graduate School of Science Kyoto University Kyoto 606-8502 Japan 2 Nikos I. Kavallaris Division of Mathematical Science Graduate School of Engineering Science Osaka University Toyonaka 560-8531 Japan We study the electrostatic MEMS-device equation, ut u = |x| (1u)p , with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions. Introduction The purpose of the present paper is to study the global-in-time behaviour of the solution to the parabolic problem ut u = u = 0 (1 u)p u|t=0 = u0(x) 0 u < 1 where > 0 is a constant, p > 1, f (x) 0, f (x) 0 is a continuous function, Rn is a bounded domain with smooth boundary , and u0 = u0(x) [0, 1) is a continuous function. This equation models the dynamic deflection of an elastic mebrane inside a micro-electro mechanical system (MEMS). This kind of systems combine electronics with various types of micro-size mechanical devices and could be found in accelerometers for airbag deployment in automobiles, in ink jet printer heads, in optical switches, in chemical sensors and so on, for more details see [21] and the references therein. The structure of the set of stationary solutions has been studied for p = 2 ([8, 11]). We obtain, similarly, an upper bound of for the existence of the solution denoted by > 0 for general p > 1. Thus, if 0 < < and > , then C = and C = , respectively, where u C2() C0() | u solves (1.1) . Also, if C = there is a unique minimal element, denoted by u. The global-in-time behaviour of the non-stationary solution has been studied also for p = 2 ([9]). More precisely, for < and > , there is a solution to (1.1) converging to u uniformly and any solution must touchdown in finite time, i.e., T < +, = 1, respectively. Here and henceforth, T > 0 denotes the existence time of the solution. We can refine these results on the global-in-time behaviour of the solution even for general p > 1. Theorem 1.1. We have the following. 1. If > , then (1.2) occurs. 3. If there is u C\{u}, u0 u, u0 u, then (1.2) occurs. Next, we take the case of = B {x Rn | |x| < 1} , f (x) = |x| , 0, and prove existence of non-minimal radially symmetric stationary solutions. To state the results, we set Cr = u C2() C0() | u = u(|x|) solves (1.1) where and p are the maximum solutions of the equations f (p, n, ) = 0 and f (p, n, 0) = 0 respectively, for f (p, n, ) = (p + 1)2 n2 4 (p + 1) (p + 3p + 1) n 4 2p + 2p (1 p) 5p2 + 2p + 1 . Note that , p are defined for n 7 and n = 7, 8, 9 respectively. ( + 2) { + 2 + (n 2)(p + 1)} (p + 1)2 In the cases of n [2, 6], n [7, 9], n 10, whenever > 0, Cr bends infinitely many times with respect to around , while at most a finite number of bendings occur to Cr when n [7, 9], n 10, The case p = 2 has also been studied on C = >0{}C. First, the above profile of Cr is observed numerically [8]. Next, some estimates on are given for nonradially symmetric case [8, 9]. Finally, existence of the non-minimal non-radially symmetric stationary solution is proven by the variational method [4]. Given u C, the linearized eigenvalue problem is defined as follows: Then, the number of its negative eigenvalues, denoted by i = i(, u), is called Morse index. In case (1.3) the number of negative eigenvalues corresponding to radially symmetric eigenfunctions, denoted by iR = iR(, u), is called radial Morse index. Theorem 1.3. Under the assumptions of Theorem 1.2, iR = iR(, u) increases one by one at each bending point. u = f (r, u) 0 in B, u = 0 on B, if fr 0, where r = | | x , see [16]. This is not the case of (1.3) with > 0. In fact, general theorems of radial symmetry of the solution to the stationary problem are not valid for > 0 ([10, 20]). There may be non-radial bifurcation and i(, u) = iR(, u) would not be always valid. This paper is composed of five sections. Theorems 1.1, 1.2, and 1.3 are proven in sections 2, 3, and 4, respectively. In the final section, 5, we confirm the role of radial Morse indices in the formation of connecting orbits. Proof of Theorem 1.1 The original proof for p = 2 to the first case of this theorem is quite technical [9], it provides with some bounds of the blow-up time though. Here we give a simpler proof by applying Kaplans method adopted by [2]. The proof of the second case is obtained by using classical techniques like maximum-principle or dynamical-systems aguments, [7, 9], and we omit it. The final case is obtained by the method of [15]. Before proceeding to these cases, we recall that Kaplans method guarantees touchdown for any > 0 if the initial value is close to 1 and f > 0 on (see also [3]). In fact, let 1 > 0 be the principal eigenvalue of , and take the solution to 1 = 11, 1 > 0 in , 1 = 0 on , normalized by 1 = 1. Then, using m = inf f > 0 and A = u1dx, we obtain via Jensens inequality (1 u)p1dx 1A + m(1 A)p. Let H(s) = 1s + m(1 s)p, s (0, 1). Then there exists a (0, 1) such that H(s) > 0 in (a, 1). Hence if we choose u0 close enough to 1 so that A(0) > a, then ddAt > 0 for all t when u is global. But, then we have t = which is a contradiction. Kaplans method could be also used to prove touchdown for big enough , see [11]. Now, we show the first case of Theorem 1.1, i.e., > implies (1.2). It suffices to assume u0 = 0 for this purpose. Then, it holds that v = 0 Then, it holds that by ut 0. This implies ut1dxds = (f1(x)u)1p dxds. (f1(x)u)1p dxds u(, t + 1)1dx w( )dx = From the monotone convergence theorem, and u(, t) 1 = |u(x, t)| dx | | < +, f(x) it follows that u(, t) has a limit w in L1() as t , while K(x, t) = (1u)p converges to K (x) = (1f(wx))p in L1( , (x)dx) as t +, where (x) = dist(x, ). Testing (1.1) by C2() with | = 0, we obtain Letting t we derive This means that w is a weak solution to w = w = 0 on , and we obtain a contradiction by the following lemma, see also [2, 9]. g(w) = h(w) = (1 s)pds and put (w) = h1(h(w)), where > 0. The following properties of this mapping w [0, 1) (w) are obtained by g > 0: (0) = 0, 0 (w) < w, (w) = (1 )g((w))g(w)1 > 0, on in weak sense, i.e., for any J C2() satisfying J 0 and J | = 0. Thus, w is a weak supersolution to (2.2) and therefore, the iteration sequence {vk}k=0 defined by vk+1 = vk+1 = 0 on with v0 = w is monotone decreasing. It also satisfies vk 0 and converges uniformly to a solution to (2.2) for (1 ) by Dinis theorem. This implies C(1) = , a contradiction by the definition of since > 0 is arbitrary. Th (...truncated)