A nonlocal quenching problem arising in a microelectro mechanical system
Authors:
JongShenq Guo, Bei Hu and ChiJen Wang
Journal:
Quart. Appl. Math. 67 (2009), 725734
MSC (2000):
Primary 35K60, 35Q72, 34B18
Published electronically:
May 14, 2009
MathSciNet review:
2588232
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, we study a nonlocal parabolic problem arising in the study of a microelectro mechanical system. The nonlocal nonlinearity involved is related to an integral over the spatial domain. We first give the structure of stationary solutions. Then we derive the convergence of a global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.
 1.
GoJien
Chen and JongShenq
Guo, Critical length for a quenching problem with nonlocal
singularity, Methods Appl. Anal. 5 (1998),
no. 2, 185–194. MR 1636562
(99i:35073)
 2.
Keng
Deng, Dynamical behavior of solutions of a semilinear heat equation
with nonlocal singularity, SIAM J. Math. Anal. 26
(1995), no. 1, 98–111. MR 1311883
(95j:35105), http://dx.doi.org/10.1137/S0036141091223881
 3.
Keng
Deng, Man
Kam Kwong, and Howard
A. Levine, The influence of nonlocal nonlinearities on the long
time behavior of solutions of Burgers’ equation, Quart. Appl.
Math. 50 (1992), no. 1, 173–200. MR 1146631
(92k:35241)
 4.
Marek
Fila and Josephus
Hulshof, A note on the quenching rate,
Proc. Amer. Math. Soc. 112 (1991),
no. 2, 473–477. MR 1055772
(92a:35090), http://dx.doi.org/10.1090/S00029939199110557727
 5.
Stathis
Filippas and JongShenq
Guo, Quenching profiles for onedimensional semilinear heat
equations, Quart. Appl. Math. 51 (1993), no. 4,
713–729. MR 1247436
(95b:35029)
 6.
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),
no. 2, 434–446 (electronic). MR 2285871
(2007k:35330), http://dx.doi.org/10.1137/060648866
 7.
Yoshikazu
Giga and Robert
V. Kohn, Asymptotically selfsimilar blowup of semilinear heat
equations, Comm. Pure Appl. Math. 38 (1985),
no. 3, 297–319. MR 784476
(86k:35065), http://dx.doi.org/10.1002/cpa.3160380304
 8.
JongShenq
Guo, On the quenching behavior of the solution of a semilinear
parabolic equation, J. Math. Anal. Appl. 151 (1990),
no. 1, 58–79. MR 1069448
(91g:35021), http://dx.doi.org/10.1016/0022247X(90)902439
 9.
JongShenq
Guo, On the quenching rate estimate, Quart. Appl. Math.
49 (1991), no. 4, 747–752. MR 1134750
(92j:35097)
 10.
JongShenq
Guo, Quenching behavior for the solution of a nonlocal semilinear
heat equation, Differential Integral Equations 13
(2000), no. 79, 1139–1148. MR 1775250
(2001f:35199)
 11.
JongShenq
Guo and TsungMin
Hwang, On the steady states of a nonlocal semilinear heat
equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.
8 (2001), no. 1, 53–68. Advances in quenching.
MR
1820665 (2002c:35135)
 12.
Hideo
Kawarada, On solutions of initialboundary problem for
𝑢_{𝑡}=𝑢ₓₓ+1/(1𝑢), Publ.
Res. Inst. Math. Sci. 10 (1974/75), no. 3,
729–736. MR 0385328
(52 #6192)
 13.
Howard
A. Levine, Quenching, nonquenching, and beyond quenching for
solution of some parabolic equations, Ann. Mat. Pura Appl. (4)
155 (1989), 243–260. MR 1042837
(91m:35028), http://dx.doi.org/10.1007/BF01765943
 14.
John
A. Pelesko, Mathematical modeling of electrostatic MEMS with
tailored dielectric properties, SIAM J. Appl. Math.
62 (2001/02), no. 3, 888–908 (electronic). MR 1897727
(2003b:74025), http://dx.doi.org/10.1137/S0036139900381079
 15.
J.
A. Pelesko and A.
A. Triolo, Nonlocal problems in MEMS device control, J. Engrg.
Math. 41 (2001), no. 4, 345–366. MR
1872152, http://dx.doi.org/10.1023/A:1012292311304
 1.
 G.J. Chen and J.S. Guo, Critical length for a quenching problem with nonlocal singularity, Methods and Applications of Analysis 5 (1998), 185194. MR 1636562 (99i:35073)
 2.
 K. Deng, Dynamical behavior of solutions of a semilinear heat equation with nonlocal singularity, SIAM J. Math. Anal. 26 (1995), 98111. MR 1311883 (95j:35105)
 3.
 K. Deng, M.K. Kwong, and H.A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers' equation, Quart. Appl. Math. 50 (1992), 173200. MR 1146631 (92k:35241)
 4.
 M. Fila and J. Hulshof, A note on the quenching rate, Proc. Amer. Math. Soc. 112 (1991), 473477. MR 1055772 (92a:35090)
 5.
 S. Filippas and J.S. Guo, Quenching profiles for onedimensional semilinear heat equations, Quart. Appl. Math. 51 (1993), 713729. MR 1247436 (95b:35029)
 6.
 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 (2007), 434446. MR 2285871 (2007k:35330)
 7.
 Y. Giga and R.V. Kohn, Asymptotically selfsimilar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297319. MR 784476 (86k:35065)
 8.
 J.S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl. 151 (1990), 5879. MR 1069448 (91g:35021)
 9.
 J.S. Guo, On the quenching rate estimate, Quart. Appl. Math. 49 (1991), 747752. MR 1134750 (92j:35097)
 10.
 J.S. Guo, Quenching behavior for the solution of a nonlocal semilinear heat equation, Differential and Integral Equations 13 (2000), 11391148. MR 1775250 (2001f:35199)
 11.
 J.S. Guo and T.M. Hwang, On the steady states of a nonlocal semilinear heat equation, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 8 (2001), 5368. MR 1820665 (2002c:35135)
 12.
 H. Kawarada, On solutions of initial boundary value problem for , RIMS Kyoto U. 10 (1975), 729736. MR 0385328 (52:6192)
 13.
 H.A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. 155 (1989), 243260. MR 1042837 (91m:35028)
 14.
 J.A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math. 62 (2002), 888908. MR 1897727 (2003b:74025)
 15.
 J.A. Pelesko, A.A. Triolo, Nonlocal problems in MEMS device control, J. Eng. Math. 41 (2001), 345366. MR 1872152
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35K60,
35Q72,
34B18
Retrieve articles in all journals
with MSC (2000):
35K60,
35Q72,
34B18
Additional Information
JongShenq Guo
Affiliation:
Department of Mathematics, National Taiwan Normal University, 88, S4, Ting Chou Road, Taipei 11677, Taiwan; and Taida Institute of Mathematical Sciences, National Taiwan University, 1, S4, Roosevelt Road, Taipei 10617 Taiwan
Bei Hu
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
ChiJen Wang
Affiliation:
Department of Mathematics, National Taiwan Normal University, 88, S4, Ting Chou Road, Taipei 11677, Taiwan
DOI:
http://dx.doi.org/10.1090/S0033569X09011595
PII:
S 0033569X(09)011595
Keywords:
Quenching,
nonlocal parabolic problem,
microelectro mechanical system
Received by editor(s):
July 23, 2008
Published electronically:
May 14, 2009
Additional Notes:
The first author was partially supported by the National Science Council of the Republic of China under the grant NSC 962119M003001.
Article copyright:
© Copyright 2009 Brown University
