A HYPERBOLIC NON-LOCAL PROBLEM MODELLING MEMS TECHNOLOGY N. I. KAVALLARIS, A. A. LACEY, C. V. NIKOLOPOULOS, AND D. E. TZANETIS

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1 A HYPERBOLIC NON-LOCAL PROBLEM MODELLING MEMS TECHNOLOGY N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS Abstract In this work we study a non-local hyperbolic equation of the form u tt = u xx + u + α, u dx with homogeneous Dirichlet boundary conditions and appropriate initial conditions The problem models an idealised electrostatically actuated MEMS Micro-Electro-Mechanical System device Initially we present the derivation of the model Then we prove local existence of solutions for > and global existence for < < for some positive, with zero initial conditions; similar results are obtained for other initial data For larger values of the parameter, ie when > + for some constant +, and with zero initial conditions, it is proved that the solution of the problem quenches in finite time; again similar results are obtained for other initial data Finally the problem is solved numerically with a finite difference scheme Various simulations of the solution of the problem are presented, illustrating the relevant theoretical results Introduction The aim is to study the problem u tt = u xx + u + α, u < x <, t >, a u, t =, u, t =, t >, b ux, = u x, u t x, = v x, < x <, c in the case where, α are positive parameters When α = problem reduces to the local hyperbolic problem u tt = u xx + / u, < x <, t >, a u, t =, u, t =, t >, b ux, = u x, u t x, = v x, < x <, c which has been studied in [3, 4, 4, ] For α > we obtain a non-local hyperbolic problem which, to our knowledge, has not been previously studied The equation describes the operation of an idealised electrostatically actuated MEMS Micro-Electro-Mechanical System device, which consists of a membrane and a rigid plate placed in a parallel position, and connected in series with a capacitor The membrane is taken to be rectangular with two fixed parallel sides, while the other sides are free A potential difference applied between the membrane and the plate causes the membrane to be deformed Date: 7th April 99 Mathematics Subject Classification Primary 35K55, 35J6; Secondary 74H35, 74G55, 74K5 Key words and phrases Electrostatic MEMS, Quenching of Solution, Hyperbolic Non-Local Problems

2 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS The deformation u satisfies equation a assuming that dissipation, which might result from either an electric current flowing through a resistance or viscous effects on the moving membrane, can be neglected The parameter α represents the ratio of a reference capacitance to the fixed capacitance, in particular, the reference will be the capacitance of the device when its displacement vanishes, while is a control parameter proportional to the square of the applied voltage A deformation of u = at some point x, t, represents the membrane having been moved so that it touches the plate the zero-potential equilibrium separation is, in dimensionless variables More specifically in the derivation of the model, [9, 4], it is possible to consider a dissipative, u t, term which can possibly dominate the acceleration, u tt, term In this case the version of equation a is parabolic: u t = u xx + u + α, < x <, t >, 3a dx u u, t =, u, t =, t >, 3b ux, = u x, < x < 3c This problem has been extensively studied [5, 6,, 4, 5], etc, although there are still a number of open questions Including both the dissipative and inertial terms we would get equation a but with an additional term ɛu t, for some constant ɛ, on the left-hand side For ɛ small, ie negligible dissipation, equation a is obtained as an approximation of the model Note that for the non-local problem there are no results regarding the behaviour of the solution However, there are some results for the local version of the problem in [3], for the one-dimensional case, and in [7] for the two- and three-dimensional cases The local parabolic problem u t = u + fx/ u, x Ω, t >, 4a ux, t =, x Ω, t >, 4b ux, = u x, x Ω, 4c where Ω is a smooth domain of R N, N, also describes the operation of a MEMS device for a simpler regime Here the positive parameter is again proportional to the square of a fixed applied voltage, while fx represents spatial variation of the electrical permittivity of the elastic membrane; by the physics of the problem, f is a non-negative function A dielectric profile which is often encountered in applications is fx = x p, p > For such a case, there exists a critical value of,, [, ], usually called pull-in voltage in MEMS literature, such that the steady-state problem corresponding to 4 has no solution for > Moreover, for such values of, the solution ux, t of 4 quenches in finite time, ie min x { ux, t} as t T <, see [3, 7, ] The quenching behaviour physically corresponds to the phenomenon of touch-down, ie the elastic membrane touches the rigid plate For a thorough study of the structure of the solutions of the steady-state problem corresponding to 4, see [7,, ], while for a further study of 4 see [8, 9] Regarding problem 3, it has also been proved that quenching occurs for large enough values of the parameter, [5, ], as well as for big enough initial datum u x, see [6] Furthermore, some estimates of the pull-in voltage as well as the study of the stability,

3 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 3 through the Morse index, of the steady states of 3 for higher dimensions are presented in [6] Note that finite time of existence has been more frequently studied for blow-up in equations such as u t = u xx + fu/ fu dx p, where now u as t T < ; such equations can model Ohmic heating or shear-band formation, see for example [] and [] The present paper is organized as follows In Section we present the derivation of the model The local existence of is studied in Section 3 The corresponding steady-state problem is studied in Section 4, while global existence of is established for small enough values of the parameter in Section 5 Section 6 is devoted to the study of the quenching of solutions of problem More precisely, quenching of solution ux, t is proved for a large enough value of the parameter first in the case of zero initial data, and second for non-zero data Finally in Section 7 a numerical solution of the problem is given via a finite difference two-step Crank-Nicolson scheme and some simulations illustrating the theoretical results of this work are presented The study of the local behaviour near quenching will be investigated in a subsequent paper, using a self-similar approach as has been done for the blow-up behaviour of standard semilinear wave equations, see [5, 6, ] Derivation of the Model We consider an idealised electrostatically actuated MEMS device It consists of a membrane and a rigid plate which are placed parallel to each other The membrane has two parallel sides fixed, while the other sides are free For simplicity we only look at the homogeneous case, f A potential difference, not a-priori known, is applied between the top surface the membrane and the rigid plate Both membrane and plate have width w and length L, and in the undeformed state for the membrane the distance between the membrane and the plate is l We assume here that the gap between the plate and membrane, typically of order l, is small, l L and l w, and is occupied by some inviscid material with dielectric constant one, so permittivity is that of free space, ɛ Taking the potential difference across the device to be V, and assuming that the plate is earthed, the small aspect ratio of the gap gives potential, φ, to leading order, φ = V l z /l u where u is the displacement of the membrane towards the plate u = l corresponds to touch-down and z is the distance measured from the undisturbed membrane position towards the plate The electrostatic force per unit area on the membrane in the z direction is then surface charge density electic field = ɛ φ z = ɛ V /l u Taking the sides of length w, say x = and x = L, to be fixed, with those of length L, say y = and y = w, to be free, there to be no variation in the y direction, so u = u x, z, t for time t, the surface density of the membrane to be ρ, and there to be a constant surface tension T m in the membrane, its displacement satisfies the forced wave equation ρu t t = T mu x x + ɛ V /l u

4 4 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS The boundary conditions u, t = u L, t = 3 also hold Partially scaling variables, u = lu, x = Lx, t = L ρ/t m t, these become and u tt = u xx + ɛ V L /T m l / u, 4 u, t = u, t = 5 Now we suppose that the MEMS device, which will have a capacitance C depending on displacement, is connected in series with a capacitor of fixed capacitance C f and a source of fixed voltage V s Then V s = Q C c = Q + C C f, where Q is the charge on the device and fixed capacitor, and C c the series capacitance of the two Thus the potential difference, V, across the MEMS device will be V s V = 6 + C/C f In addition w L Q = ɛ φ z x, y, dx dy = V wlɛ l u dx, using, so C = C u dx, where C = wlɛ /l is the capacitance of the undeflected device Substituting this into 6 and the result into 4 leads to u tt = u xx + u + α, 7 dx u with = ɛ V s L /T m l and α = wlɛ /lc f Conditions 5 continue to apply Cases of interest often have vanishing initial conditions, ux, = u t x, =, but we shall also look at cases with more general conditions u x, v x 3 Local Existence of Solutions In this section we establish local existence of the solution of problem u tt = u xx + u + α, < x <, t > 3a dx u u, t =, u, t =, t >, 3b ux, = u x, u t x, = v x, < x < 3c where u, v C, and u = u =, by modifying appropriately the proof given for the local problem in [3], see also [] Definition 3 We say that u is a weak solution of 3 in Q T,, T if:

5 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 5 i u is continuous in Q T and satisfies the initial and boundary conditions ii There exists some δ > such that u δ in Q T iii u has weak derivatives u x, u t in Q T and for all t, T, u x, u t L, iv For any function ζx, t C Q T satisfying the boundary conditions and for t T, ζx, tu t x, t dx = + t t [ζ τ x, τu τ x, τ ζ x x, τu x x, τ] dx dτ ζx, τ dx dτ ux, τ + α 3 dy uy,τ v The total energy associated with 3 is preserved see also Section 5, ie E T t = u x + u t dx + α + α = E T := E 33 dx u We consider a fixed δ, and assume that the initial data satisfy the condition u + T v < δ 34 for a positive T Define the odd periodic with period two extensions with respect to x on R [, T ] of u, u, v which are denoted, without any confusion, again as u, u, v We also define the function G p : R [, +, R, as { F u, x [n, n +, G p x, t, u = 35 F u, x [n, n, for n =, ±, ±,, where F u = u + α dx u Then by standard arguments applied to wave equations, we have that u is a solution of 3 if and only if u solves in R [, T ] the integral equation where ux, t = u x, t + Note that 34 implies that t x+t+τ x t+τ u x, t = [u x + t + u x t] + G p y, τ, uy, τ dy dτ, 36 x+t x t v z dz u T = sup u t < δ 37 t [,T ] Let B T be the Banach space of continuous odd periodic with period two functions with respect to x, defined in R [, T ], vanishing on x = n, n Z, with the norm T Also

6 6 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS let Bu, δ be the closed ball of radius δ centred on u in B T Consider the operator T [ux, t] = u x, t + t x+t+τ x t+τ G p y, τ, uy, τ dy dτ, which, due to the definition of the function G p, is well-defined In order to show that 3, or equivalently 36, has a local-in-time solution, it is enough to show that the operator T is a contraction from Bu, δ to Bu, δ Note that u B T In particular, we have to show that T u u T < δ, T u T v T < K u v T, for u, v Bu, δ and < K < In the following, for convenience, we write fu := and Iu := fudx, hence u F u := f u Then we have [+αiu] T vx, t T ux, t T = t x+t+τ [G p y, τ, vy, τ G p y, τ, uy, τ] dy dτ, T where for x t + τ > x t+τ [G p y, τ, vy, τ G p y, τ, uy, τ] = = f v [ + αiv] [ + αiu] + f v [ + αiv] f u [ + αiu] [ + αiu] f v f u Note that f v f u = fv + fufu fv and fu fv T f δ u v T, by taking into account 37 and the fact that f is a convex function Moreover we have [ + αiv] [ + αiu] = [ + αiu] [ + αiv] [ + αiv] [ + αiv] < α[ + αiu + Iv][Iu Iv] since Iu, Iv > Using again 37 and the fact that f is an increasing and convex function we derive Iu Iv T f δ u v T and + αiu + Iv T [ + αf δ] Thus finally Gp y, τ, vy, τ G p y, τ, uy, τ αf δf δ[ + αf δ] u v T or +f δf δ u v T, Gp y, τ, vy, τ G p y, τ, uy, τ Cf, δ u v T, for Cf, δ = αf δf δ[ + αf δ] + f δf δ The same final estimate also holds when x t + τ < Therefore T vx, t T ux, t T = t x+t+τ [ Gp y, τ, vy, τ G p y, τ, uy, τ ] dydτ T x t+τ t Cf, δ u v T σ Cf, δ u v T, 38 for t σ Note that K = σ Cf, δ < if σ < Cf,δ

7 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 7 It remains to show that T u u T < δ We have again for x t + τ >, T ux, t u T = t x+t+τ G p y, τ, uy, τdy dτ x t+τ T t x+t+τ f u x t+τ [ + αiu] dy dτ T t x+t+τ f δ dydτ T x t+τ t f δ σ f δ Since the same final estimate is obtained for x t + τ < when the first inequality in 38 becomes strict, we end up with T ux, t u T δ as long as σ f δ δ, δ ie if σ f δ Thus finally, if we choose σ such that { σ < min T, δ f δ, Cf, δ }, we conclude that the operator T : Bu, δ Bu, δ is a contraction and hence the Banach fixed point theorem guarantees the existence of a unique fixed point for T We have thus established local existence of solution of problem in an interval [, σ] In particular: Theorem 3 If the initial data u x, v x C, satisfy condition 34, then, for any >, problem 3 has a unique weak C solution on Q T =, [, T ] if T is sufficiently small Remark 33 By the proof of the above theorem we also obtain that the solution u to 3 is piecewise C in Q T Furthermore, the solution of 3 could be extended to any interval of the form [, T + τ] for τ sufficiently small and positive as long as u < on Q T Remark 34 In the case of zero initial datum we could obtain, by differentiating relation 36, that ux, t is a regular solution to 3 except on the point set see also [3] {x, t R [, T ] x, x t, x + t are integers}, Definition 35 The solution ux, t of problem quenches at point x, in finite time < T < if there exist sequences {x n } n=, and {t n } n=, with x n x, t n T and ux n, t n as n In the case where T = we say that ux, t quenches in infinite time at x By Remark 33 we conclude that the solution ux, t to problem ceases to exist only by quenching In MEMS terminology quenching is usually called touch-down since it describes the phenomenon when the elastic membrane touches the rigid plate on the bottom of the MEMS device In the physical problem this is usually followed by destruction of the MEMS device The study of the quenching phenomenon is also important from mathematical point of view, because when it occurs there is singular behaviour of ux, t

8 8 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS 4 The Steady State The steady-state problem corresponding to 3 is w + w + α =, < x < ; w =, w = 4 dx w If we set W = w then 4 becomes where W = µ/w, < x < ; W =, W =, 4 / µ = + α W dx 43 Then multiplying both sides by W and integrating from m = min{w x, x [, ]} = W / to W we derive W x x W dw = W W W W dx = µ W dx = µ dw m W, hence This gives equivalently which implies x = m µ W = µ m W dx m W dw = µ W m, [ W W m m lnm + m ln W + W m ] This yields, on setting x = so that W =, [ µ = m m m lnm + m ln + m ] Furthermore we have W dx = dx dw m dw W = dw µ m W W m = m m lnm + m ln + m dw m W W m m + m = m m lnm + m ln + m ln m On using 43, we finally establish the following relation between and m, [ ] + m m + m = m m + m ln + α ln 44 m m From the above relation, we can obtain the response bifurcation diagram of problem 4, see Figure for the case of α = In particular, 44 implies that α mln m as m

9 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 9 Note that the numerical results in Section 7 indicate that the long-time behaviour of the solution, when it exists for all times, is periodic and the relevant steady-state solution is, from stability point of view, a centre This means that although we have existence of steady-state solutions for every <, it may happen that for quite close but less than the solution of the evolutionary problem could oscillate around the steady solution and approach, ie it retains significant deformation and speed, and therefore it is more likely to quench In conclusion, the critical value for the parameter, say, for which the solution of the hyperbolic problem exists for all times with vanishing initial data, is expected to be lower than For a thorough study of the structure of the set of the steady-state solutions to in higher dimensions see [6] wx Figure Bifurcation diagram for problem 4 with α = Note that for this case, Global Existence In order to prove global existence of problem, we need as a first step to determine the corresponding energy functional, see also Section 3 Unless otherwise stated, now denotes To find the energy, we multiply equation a by u t and integrate over [, ] Thus we obtain u t u tt u t u xx dx = u t dx u + α, u dx

10 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS or d dt Thus finally we get u t + u x dx = d dt = d α dt u t + ux dx + α + α u dx + α dx u + α dx u u dx = E, 5 where E is a constant representing the initial energy of the system which is conserved and is given by E = v + u x dx + α + α 5 u dx From equation 5 we deduce that u t + u x + α + α where u t = u t, t, u x = u x, t, and that dx u = E, u x + α + α dx E 53 u At this point we can use the one-dimensional Sobolev embedding and the Poincaré inequality to obtain 4u x, t u x, for x, t [, ] [, T ] 54 We also define M, M := max{ux, t, x, t [, ] [, T ]} For cases where M < irrespective of the value of T, the maximal time of existence of u is infinite Therefore we have that u M, u M, + α dx + α, and that u M + α dx + α u M Thus we obtain by 53 u x + E α + α/ M and due to 54 we conclude that 4u x, t + E for x, t [, ] [, T ] 55 α + α/ M Considering the simplest case where u =, v = we have that E = /αα + For simplicity in the following we take α = so E = / Note that the solution u ceases to exist if M =, while for α = the inequality 55 yields hm = hm ; := M + M/ M E 56

11 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION We have that h = and h =, hence for < 4 we have that h > h Also h M = 4M / M with h = 4 > for < 4 Hence we have the following global-in-time existence result Theorem 5 Problem with zero initial data has a global-in-time solution, ie < ux, t < for all x, t [, ] [,, provided that < 4 With the largest possible for every < to give a global solution of, in other words is the infimum of those which produce quenching, 4 Now consider the case of non-zero initial data, ie u or v In this case the initial energy is given by the expression E = E, u, v = v + u x + α + α u dx Following the above, for u to exist it would then be enough to show that 4M + M v + u x + α + α M α F u, 57 for F u = / + α u dx, guarantees that M < Clearly inequality 57 fails at M = if 4 > v + u x + /α 58 since F u Thus the solution exists for all time as long as < α v + u x NB With u =, F u = /+α, and a stronger estimate, as of Thm 5, is achieved 6 Quenching 6 Quenching for zero initial conditions In the first part of this section we impose zero initial conditions, ie we consider the following problem u tt = u xx + u + α, < x <, t >, 6a dx u u, t =, u, t =, t >, 6b ux, =, u t x, =, < x < 6c Our purpose is to show that there exists a critical value + of such that the solution of problem 6 quenches in finite time for any + Thus + is the supremum of those for which 6 has a global solution For convenience and due to the expected symmetry of the solution ux, t at the point x = we will consider the equivalent problem u tt = u xx + u + α, < x < dx, t >, 6a u u, t =, u x, t =, t >, ux, =, u t x, =, < x < 6b We restrict our analysis for times < t < / so that the characteristic line t = x remains in the strip < x < / Because 6 lacks clear monotonicity properties, it is not obvious that + =

12 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS Initially we consider the general problem u tt u xx = gt, < x <, t >, 63a u, t =, u x, t =, t >, ux, =, u t x, =, < x <, 63b with gt > and continuous For x t, the solution of 63 is given by ux, t = = t x+t s t x t s gs dξ ds = t gt [x + t s x + t s] ds t sgs ds := Gt, 64 since in that case the domain of dependence of the point x, t is the triangle D = {ξ, s x t s < ξ < x + t s and < s < t} On the other hand, for x t we have to subtract the contribution coming from x < so we get ux, t = Gt Gt x = t t sgs ds t x t x sgs ds 65 Note that in this case the domain of dependence is no longer a triangular but is instead split into two parts D = {ξ, s x t s < ξ < x + t s and t x < s < t} and D = {ξ, s t x s < ξ < x + t s and < s < t x} From the above analysis we bear in mind that the solution of 63 in the area x t is only a function of time, ux, t = Gt In addition for t / by the form of the solution we can easily deduce that u x = for x t, = t x gs ds > for x t Finally for every < x < /, < t < / we have ux, t Gt = max x [,/] ux, t In the following we consider the more general problem u tt u xx = hx, t, < x <, t >, 66a u, t =, u x, t =, t >, ux, =, u t x, =, < x <, 66b with hx, t >, h x x, t > and continuous Using the same reasoning as above we easily obtain that the solution of problem 66 is given by for x t, and ux, t = t t x ux, t = x+t s x t s t x+t s x t s hy, s dξ ds + hy, s dξ ds, 67 t x x+t s t s x hy, s dξ ds, for x t Due to the fact that hx, t > and h x x, t > we have u x x, t and hence max x [,/] ux, t is given by 67 In our case, going back to our original problem 6, the function h has the specific form hx, t = fugt where fu = u and gt = +α R / Note also that the u dx function f is such that f = and f s > for < s < We shall show that for x t the solution of problem 6 is purely a function of time, ie ux, t = Ut = max x [,/] ux, t, by applying a Picard iteration

13 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 3 Initially we define u to be the solution of the problem u tt u xx = gt, < x <, < t <, with the boundary and initial conditions as defined above As we have already stated, by the analysis of the problem 63, u x, t = U t for x t, u x and u x, t U t where U t = max x [,/] u x, t coincides with Gt given by 64 We inductively define the function u n x, t to be the solution of the problem u ntt u nxx = fu n gt, < x <, < t <, with the standard initial and boundary conditions By the analysis of problem 66 we have, since hx, t = fu n gt > and h x x, t = f u n u n x gt > f > and u n x > by the induction hypothesis, that u n has the required property u nx > By the induction hypothesis we also have that u n x, t is purely a function of time in the area x t hence fu n gt is also, implying that u n x, t = U n t = max x [,/] u n x, t for any n N and x t and U n t is given by U n t = t x+t s x t s fu n sgs dξ ds, 68 which implies that the sequence {U n t} n= is increasing, recalling that fs > and f s > for < s < Moreover, we have that < U n t < for every < t < t for some t / hence {U n t} n= converges as n to a function Ut which is actually the maximum value of the unique solution ux, t of 6 achieved for x t, ie ux, t = Ut = max x [,/] ux, t for x t The latter implies that the solution of problem 6 satisfies for x t the equation and since U tt = fugt, gt / [ + α/ U], Ut finally satisfies the differential inequality U tt / [ + α U], 69 with U = and U t = Therefore we obtain that the following inequality is satisfied [ ] 3 U U + α U t + α sin +, + α + α and we deduce that Ut reaches before time t = /, provided that [ + α + α sin + α] = + α 6 + α Finally we have proved the following result: Theorem 6 If + α, where + α is given by 6, then the solution ux, t to problem 6 quenches in finite time T /, ie u, t as t T / The least upper bound for global existence, +, then satisfies + + α Remark 6 Note that for α = we have that the above relation gives + =

14 4 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS Remark 63 For the local problem, ie when α =, differential inequality 69 implies that u tt, t blows up at the quenching time, ie u tt, t + as t T, which is in agreement with the result in [4] A result similar to Theorem 6 can be deduced when the initial data are non-zero and this is the subject of the next subsection 6 Quenching depending on the initial data We now prove quenching for large enough, depending on the initial data u and v, using some of the arguments developed in the previous subsection We consider problem and we assume that u is bounded away from one Let w be the solution of the problem w tt w xx =, < x <, t >, 6a w, t = w, t =, t >, wx, = u x, w t x, = v x, < x < 6b We also define the function v as the difference v = u w hence vx, = and v t x, = ; moreover v satisfies homogeneous Dirichlet boundary conditions v, t = = v, t Now we assume that u x and v x are smooth enough so that w t x, t is bounded below for x /4, 3/4 and t, /4 Then for some t < /4 there exists ɛ > such that sup /4,3/4,t wx, t < ɛ Note that t and ɛ can be taken independent of, and, having fixed t, ɛ can be arbitrarily small We define C, independent of and of ɛ, by C = inf [,] [,t ]{w} Now take t ɛ, t such that v, t < ɛ for t < t Then Hence, for < t < t, ɛ < u < + C ɛ, C ɛ u = w + v ɛ + ɛ for t t + α/ + C ɛ < + α and / + C ɛ + α/ɛ < v tt v xx < / ɛ u dx < + α/ɛ [ ] α + + C ɛ In particular, following the previous subsection, since t x t for /4 x 3/4, t /4, for < t < t t /4, where vx, t = V t for 4 x 3 4, with vx, t V t for x 4 and for x 3 4, / [ ] < / + C ɛ + α/ɛ < d V/dt α < ɛ + + C ɛ Then, at t = t ɛ,, v t = dv dt t + C ɛ + α/ɛ and for /4 < x < 3/4 t + C ɛ + α/ɛ < v = V < t [ ], ɛ + α +C ɛ

15 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 5, Additionally, v V ɛ is guaranteed in t t by taking t ɛ [ 3 + α +C ɛ] for example by [ ] α t = + ɛ C ɛ In t < t < t, /4 x 3/4, we still have vx, t = V t with d V/dt >, as long as the solutions u and v exist It follows that v, t t = V t > + C ɛ + α/ɛ + t t t + C ɛ + α/ɛ Assuming the solution exists up to t, ie T t, u, t = v, t + w, t t t t / > C + + C ɛ + α/ɛ if t t t / + C ɛ + α/ɛ 63 Therefore, taking t to be given by 6, quenching before t = t will be guaranteed on choosing large enough to satisfy [ ] α t t + ɛ 3 / t + C ɛ + α/ɛ 64 + C ɛ Note that, for restricted regions of smoothness for the initial data, different intervals of x and smaller values of t could be used 7 Numerical Solution We now carry out a brief numerical study of problem, with a variety of initial conditions A three-step Crank-Nicolson scheme Richtmayer s method is used, indicating unconditional stability Taking a partition of M + points in [, ], = x, x + δx = x,, x M = and using a time step δt for a partition in the time interval [, T ], with t i = iδt, i =,, N = T δt, we have for the approximation of the solution u, u i j ut i, x j the following scheme u i+ j u i j + u i j = u i+ j+ ui+ j + u i+ j δt + δx + u i j + α u i j+ ui j + u i j δx 7 dx u i The integral dx, where u i = u i u, u i,, u i i M T, is evaluated in each time step by Simpson s rule Taking into account the Dirichlet boundary conditions, the numerical scheme takes the form R ui+ j + + R u i+ j R ui+ j+ = R ui j + R u i j + R ui j+ for j =,, M, + δt F u i j, 7 + R u i+ R ui+ = + R u i + R ui + δt F u i,

16 6 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS for j =, R ui+ M + + R u i+ M = R ui M + R u i M + δt F u i M, for j = M, where R = δt δx, ui =, u i M = and F u i j = This results in the algebraic system or u i j + α 73 dx u i Au i+ = Bu i bu i u i+ = A Bu i A bu i, where A, B are M M matrices and b is a M vector =4 8 ux,t t 5 4 x 6 8 Figure The numerical solution of problem against space and time for = 4 In the figures of this section, the results of various numerical simulations are presented The parameters that were used are α =, M =, and R = 5 In the first two figures, = 4; this value is chosen to be slightly greater than the value 4 found, Theorem 5, to guarantee global existence In Figure, ux, t is plotted against x [, ] and t [, T ] for T = 5 In the next plot, Figure 3, ux, t is plotted against x for various times The uppermost line corresponds to the solution of the problem near quenching These numerical results indicate that for zero initial data, 4 and suggest that Theorem 5 is close to best possible The remaining three figures show solutions for = < 4 In Figure 4, we consider a situation where the solution does not quench but instead oscillates Additionally, in Figure 5, the solution at the mid point, x = /, is plotted against time Finally, Figure 6 shows quenching of the solution for large enough initial data In this example, u =, v =, still with =

17 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 7 4 =4 ux,t x Figure 3 Profile of the numerical solution of problem 6, for various times and for = 4 = 5 ux,t t 4 x 6 8 Figure 4 The numerical solution of problem 6, for = References [] JW Bebernes and AA Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv Diff Eqns,, 997, [] JW Bebernes and AA Lacey, Shear Band Formation for a Non-Local Model of Thermo-Viscoelastic Flows, Adv Math Sci Appl, 5, 5, 65-8 [3] PH Chang and HA Levine, The Quenching of Solutions of Semilinear Hyperbolic Equations, SIAM Journal of Mathematical Analysis,, 6 98 pp [4] CY Chan and KK Nip, On the blow-up of u tt at quenching for semilinear Euler-Poisson-Darboux equations, Comp Appl Mat, 4, 995, 85 9 [5] P Bizon, T Chmaj and Z Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 7, 4, 87 [6] P Bizon, D Maison and A Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity,, 7, 6 74 [7] P Esposito, N Ghoussoub and Y Guo, Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity, Comm Pure Appl Math 6 7,

18 8 N I KAVALLARIS, A A LACEY, C V NIKOLOPOULOS, AND D E TZANETIS 4 = 8 u/t t Figure 5 The numerical solution of problem 6, u/, t is plotted for = =, u =, v = 8 ux,t t 4 x 6 8 Figure 6 The numerical solution of problem 6, ux, t is plotted against space for u =, v = and = [8] P Esposito and N Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl Anal 5 8, [9] G Flores, G Mercado, JA Pelesko and N Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J Appl Math 67 7, [] VA Galaktionov and SI Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation, Quart Appl Math 6 3, [] PR Garabedian, Partial Differential Equations, John Wiley, New York, 964 [] N Ghoussoub and Y Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J Math Anal 38 7, [3] N Ghoussoub and Y Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case, Nonlinear Diff Eqns Appl 5 8, 5 45 [4] FK N Gohisse and Th K Boni, Quenching time of some nonlinear wave equations, Arch Mat 45, 9, 5-4 [5] J-S Guo, B Hu and C-J Wang, A non-local quenching problem arising in micro-electro mechanical systems, Quarterly Appl Math to appear

19 ON A NON-LOCAL HYPERBOLIC MEMS EQUATION 9 [6] J-S Guo and NI Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control, preprint [7] Y Guo, Z Pan, and MJ Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM JAppl Math 66 6, [8] Y Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, J Diff Eqns, 44 8, [9] Y Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J Diff Eqns, 45 8, [] KM Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, arxiv:8949v [mathap] [] NI Kavallaris, T Miyasita, and T Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Diff Eqns Appl 5 8, [] HA Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann Mat Pura Appl , 43 6 [3] HA Levine and MW Smiley, Abstract wave equations with a singular nonlinear forcing term, J Math Anal Appl 3, [4] JA Pelesko and AA Triolo, Non-local problems in MEMS device control, J Engrg Math, 4, [5] JA Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties SIAM Journal of Applied Mathematics, 6, 3 pp [6] JA Pelesko and DH Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, [7] RA Smith, On A Hyperbolic Quenching Problem In Several Dimensions, SIAM Journal of Mathematical Analysis,, pp 8 94 Department of Statistics and Actuarial-Financial Mathematics, University of Aegean, Building B, TGr-83 Karlovassi, Samos, Greece address: nkaval@aegeangr Department of Mathematics and Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh EH4 4AS, UK address: andrewl@mahwacuk Department of Mathematics, University of Aegean, Gr-83 Karlovassi, Samos, Greece address: cnikolo@aegeangr Department of Mathematics, School of Applied Mathematical and Physical Sciences,, National Technical University of Athens, Zografou Campus, 57 8 Athens, Greece address: dtzan@mathntuagr