American Institute of Mathematical Sciences

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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

Jong-Shenq Guo 1, and  Bo-Chih Huang 2,

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.
Keywords: quenching of solutions, nonlocal hyperbolic problem, micro-electro mechanical system., Wave equations.
Mathematics Subject Classification: Primary: 35L71, 35L81; Secondary: 35A01, 35L20, 74H35, 74K1.
Citation: Jong-Shenq Guo, Bo-Chih Huang. Hyperbolic quenching problem with damping in the micro-electro mechanical system device. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

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