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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 |
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 (): 729.
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 (): 729.
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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