Abstract

In this paper, we consider the following nonlocal model

The model is a well-known one to study micro-electromechanical systems (MEMS) devices. We prove the existence of touchdown solutions in the genera cases that  is an arbitrary bounded domain and ,  are positive numbers and f is not constant. 

References

[1] J.A. Pelesko and A.A. Triolo, “Nonlocal problems in MEMS device control, Journal of Engineering Mathematics”, pp. 345-366, 2001.
[2] N. I. Kavallaris and T. Suzuki, “Non-local partial differential equations for engineering and biology”, Mathematics for Industry (Tokyo): Springer, Cham, 2018, p. xix+300.
[3] Jong-Shenq Gou and Bei Hu, “Quenching rate for a nonlocal problem arising in the micro-electro mechanical system”, Journal of Differential Equations, pp. 3285-3311, 2018.
[4] Jong-Shenq Gou và Philippe Souple, “No touchdown at zero points of the permittivity profile for the MEMS problem”, Siam J. Math. Anal, pp. 614-625, 2015.
[5] Jong-Shenq Gou and N. I. Kavallaris, “On a nonlocal parabolic problem arising in electrostatic MEMS control, Discrete and Continuous dynamical systems”, pp. 1723-1746, 2012.
[6] Jong-Shenq Gou, Bei Hu and C. J. Wang, “A nonlocal quenching problem arising in a micro-electro mechanical system”, Quart. Appl. Math., p. 25–734, 2009.
[7] P. Esposito and N. Ghoussoub, “Uniqueness of solutions for an elliptic equation modeling MEMS”, Methods Appl. Anal., 15
[8] N. Ghoussoub and Y. Guo, “On the partial differential equations of electrostatic MEMS devices: Stationary case”, SIAM J. Math. Anal., 38 (2006/07), 1423{1449.
[9] N. Ghoussoub and Y. Guo, “On the partial differential equations of electrostatic MEMS devices. II: Dynamic case”, NoDEA Nonlinear Di_. Eqns. Appl., 15 (2008), 115{145.
[10] Y. Guo, Z. Pan and M. J. Ward, “Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties”, SIAM J. Appl. Math., 66 (2005), 309{338.
[11] Y. Guo, “On the partial differential equations of electrostatic MEMS devices. III: Refined touchdown behavior”, J. Diff. Eqns., 244 (2008), 2277{2309.
[12] Y. Guo, “Global solutions of singular parabolic equations arising from electrostatic MEMS”, J.Diff. Eqns., 245 (2008), 809{844.
[13] Z. Guo and J. Wei, “Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity”, Comm. Pure Appl. Anal., 7 (2008), 765{786.
[14] G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, “Analysis of the dynamics and touchdown in a model of electrostatic MEMS”, SIAM J. Appl. Math., 67 (2006/07), 434{446.
[15] N. I. Kavallaris, T. Miyasita and T. Suzuki, “Touchdown and related problems in electrostatic MEMS device equation”, NoDEA Nonlinear Di_. Eqns. Appl., 15 (2008), 363{385.
[16] N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, “A hyperbolic non-local problem modelling MEMS technology”, Rocky Mountain J. Math., 41 (2011), 505{534.
[17] J. A. Pelesko and D. H. Bernstein, “Modeling MEMS and NEMS”, Chapman & Hall/CRC,Boca Raton, FL, 2003.
[18] J. S. Guo, “Recent developments on a nonlocal problem arising in the micro-electro mechanical system”, Tamkang J. Math. 45 (2014) 229–241
[19] M. Fila, J. Hulshof, “A note on the quenching rate”, Proc. Amer. Math. Soc. 112 (1991) 473–477.
[20] G. Flores, G. Mercado, J.A. Pelesko, N. Smyth, “Analysis of the dynamics and touchdown in a model of electrostatic MEMS”, SIAM J. Appl. Math. 67 (2007) 434–446.
[21] M.R. Boyd, S.B. Crary and M.D. Giles, “A heuristic approach to the electromechanical modeling of MEMS beams”, Technical Digest Solid-State Sensor and Actuator Workshop (1994) pp. 123–126.
[22] D. Bernstein, P. Guidotti and J.A. Pelesko, “Analytical and numerical analysis of electrostatically actuated mems devices”, Proceeding of Modeling and Simulation of Microsystems 2000 (2000) pp. 489–492.
[23] M.T.A. Saif, B.E. Alaca and H. Sehitoglu, “Analytical modeling of electrostatic membrane actuator micro pumps”, IEEE J. Microelectromech. Syst. 8 (1999) 335–344
[24] J.-S. Guo, B.-C. Huang, “Hyperbolic quenching problemwith damping in the micro-electromechanical system device”, Discrete Contin. Dyn. Syst. Series B, 19 (2014), 419-434.
[25] H.A. Levine, “Quenching, nonquenching, and beyond quenching for solution of some parabolic equations”, Ann.Mat. Pura Appl., 155 (1989), 243–260.