March  2014, 19(2): 419-434. doi: 10.3934/dcdsb.2014.19.419

Hyperbolic quenching problem with damping in the micro-electro mechanical system device

1. 

Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137

2. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan

Received  May 2013 Revised  August 2013 Published  February 2014

We study the initial boundary value problem for the damped hyperbolic equation arising in the micro-electro mechanical system device with local or nonlocal singular nonlinearity. For both cases, we provide some criteria for quenching and global existence of the solution. We also derive the existence of the quenching curve for the corresponding Cauchy problem with local source.
Citation: Jong-Shenq Guo, Bo-Chih Huang. Hyperbolic quenching problem with damping in the micro-electro mechanical system device. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 419-434. doi: 10.3934/dcdsb.2014.19.419
References:
[1]

K. Agre and M. A. Rammaha, Quenching and non-quenching for nonlinear wave equations with damping, Canad. Appl. Math. Quart., 9 (2001), 203-223.

[2]

A. Andrew and W. Wolfgang, The quenching problem for nonlinear parabolic differential equations, in Ordinary and Partial Differential Equations, Lecture Notes in Math., 564, Springer, Berlin, (1976), 1-12.

[3]

A. Andrew and W. Wolfgang, On the global existence of solutions of parabolic differential equations with a singular nonlinear term, Nonlinear Anal., 2 (1978), 499-504. doi: 10.1016/0362-546X(78)90057-3.

[4]

L. A. Caffarelli and A. Friedman, Differentiability of the blow-up curve for one-dimensional nonlinear wave equations, Arch. Rational Mech. Anal., 91 (1985), 83-98. doi: 10.1007/BF00280224.

[5]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241. doi: 10.1090/S0002-9947-1986-0849476-3.

[6]

P. H. Chang and H. A. Levine, The quenching of solutions of semiliear hyperbolic equations, SIAM J. Math. Anal., 12 (1981), 893-903. doi: 10.1137/0512075.

[7]

S. Filippas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713-729.

[8]

J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl., 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9.

[9]

J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system, Quarterly Appl. Math., 67 (2009), 725-734.

[10]

J.-S. Guo and N. I. Kavallaris, On a nonlocal parabolic problem arising in electrostatic MEMS control, Discrete Contin. Dyn. Syst., 32 (2012), 1723-1746. doi: 10.3934/dcds.2012.32.1723.

[11]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. doi: 10.1007/s00030-007-6004-1.

[12]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.

[13]

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. doi: 10.1216/RMJ-2011-41-2-505.

[14]

H. Kawarada, On solutions of initial boundary value problem for $u_t=u_{x x}+1/(1-u)$, RIMS. Kyoto Univ., 10 (1974/75), 729-736. doi: 10.2977/prims/1195191889.

[15]

H. A. Levine, The phenomenon of quenching: A survey, in Trends in the theory and practice of nonlinear analysis, North-Holland Math. Stud., 110, North-Holland, Amsterdam, (1985), 275-286. doi: 10.1016/S0304-0208(08)72720-8.

[16]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl., 155 (1989), 243-260. doi: 10.1007/BF01765943.

[17]

H. A. Levine and J. T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal., 11 (1980), 842-847. doi: 10.1137/0511075.

[18]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147-1164. doi: 10.1353/ajm.2003.0033.

[19]

F. Merle and H. Zaag, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., 331 (2005), 395-416. doi: 10.1007/s00208-004-0587-1.

[20]

F. Merle and H. Zaag, Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math., 135 (2011), 353-373. doi: 10.1016/j.bulsci.2011.03.001.

[21]

R. A. Smith, On a hyperbolic quenching problem in several dimensions, SIAM J. Math. Anal., 20 (1989), 1081-1094. doi: 10.1137/0520072.

[22]

J. Zhu, Quenching of solutions of nonlinear hyperbolic equations with damping, in Differential and difference equations and applications, Hindawi Publ. Corp., New York, (2006), 1187-1194.

show all references

References:
[1]

K. Agre and M. A. Rammaha, Quenching and non-quenching for nonlinear wave equations with damping, Canad. Appl. Math. Quart., 9 (2001), 203-223.

[2]

A. Andrew and W. Wolfgang, The quenching problem for nonlinear parabolic differential equations, in Ordinary and Partial Differential Equations, Lecture Notes in Math., 564, Springer, Berlin, (1976), 1-12.

[3]

A. Andrew and W. Wolfgang, On the global existence of solutions of parabolic differential equations with a singular nonlinear term, Nonlinear Anal., 2 (1978), 499-504. doi: 10.1016/0362-546X(78)90057-3.

[4]

L. A. Caffarelli and A. Friedman, Differentiability of the blow-up curve for one-dimensional nonlinear wave equations, Arch. Rational Mech. Anal., 91 (1985), 83-98. doi: 10.1007/BF00280224.

[5]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241. doi: 10.1090/S0002-9947-1986-0849476-3.

[6]

P. H. Chang and H. A. Levine, The quenching of solutions of semiliear hyperbolic equations, SIAM J. Math. Anal., 12 (1981), 893-903. doi: 10.1137/0512075.

[7]

S. Filippas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713-729.

[8]

J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl., 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9.

[9]

J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system, Quarterly Appl. Math., 67 (2009), 725-734.

[10]

J.-S. Guo and N. I. Kavallaris, On a nonlocal parabolic problem arising in electrostatic MEMS control, Discrete Contin. Dyn. Syst., 32 (2012), 1723-1746. doi: 10.3934/dcds.2012.32.1723.

[11]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. doi: 10.1007/s00030-007-6004-1.

[12]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.

[13]

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. doi: 10.1216/RMJ-2011-41-2-505.

[14]

H. Kawarada, On solutions of initial boundary value problem for $u_t=u_{x x}+1/(1-u)$, RIMS. Kyoto Univ., 10 (1974/75), 729-736. doi: 10.2977/prims/1195191889.

[15]

H. A. Levine, The phenomenon of quenching: A survey, in Trends in the theory and practice of nonlinear analysis, North-Holland Math. Stud., 110, North-Holland, Amsterdam, (1985), 275-286. doi: 10.1016/S0304-0208(08)72720-8.

[16]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl., 155 (1989), 243-260. doi: 10.1007/BF01765943.

[17]

H. A. Levine and J. T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal., 11 (1980), 842-847. doi: 10.1137/0511075.

[18]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147-1164. doi: 10.1353/ajm.2003.0033.

[19]

F. Merle and H. Zaag, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., 331 (2005), 395-416. doi: 10.1007/s00208-004-0587-1.

[20]

F. Merle and H. Zaag, Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math., 135 (2011), 353-373. doi: 10.1016/j.bulsci.2011.03.001.

[21]

R. A. Smith, On a hyperbolic quenching problem in several dimensions, SIAM J. Math. Anal., 20 (1989), 1081-1094. doi: 10.1137/0520072.

[22]

J. Zhu, Quenching of solutions of nonlinear hyperbolic equations with damping, in Differential and difference equations and applications, Hindawi Publ. Corp., New York, (2006), 1187-1194.

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