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On the blow-up results for a class of strongly perturbed semilinear heat equations
On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models
1. | School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
2. | School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China |
References:
[1] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities,, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63.
|
[2] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation,, J. Differential Equations, 243 (2007), 208.
doi: 10.1016/j.jde.2007.05.031. |
[3] |
N. D. Brubaker and A. E. Lindsay, Analysis of the singular solution branch of a prescribed mean curvature equation with singular nonlinearity modeling a MEMS capacitor,, Preprint, (). Google Scholar |
[4] |
N. D. Brubaker and A. E. Lindsay, The onset of multivalued solutions of a prescribed mean curvature equation with singular nonlinearity,, Eur. J. Appl. Math., 24 (2013), 631.
doi: 10.1017/S0956792513000077. |
[5] |
N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models,, Eur. J. Appl. Math., 22 (2011), 455.
doi: 10.1017/S0956792511000180. |
[6] |
N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity,, Nonlinear Anal., 75 (2012), 5086.
doi: 10.1016/j.na.2012.04.025. |
[7] |
N. D. Brubaker, J. I. Siddique, E. Sabo, R. Deaton and J. A. Pelesko, Refinements to the study of electrostatic deflections: theory and experiment,, Eur. J. Appl. Math., 24 (2013), 343.
doi: 10.1017/S0956792512000435. |
[8] |
M. Burns and M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux,, Eur. J. Appl. Math., 22 (2011), 317.
doi: 10.1017/S0956792511000076. |
[9] |
Y.-H. Cheng, K.-C. Hung and S.-H. Wang, Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS,, Nonlinear Anal., 89 (2013), 284.
doi: 10.1016/j.na.2013.04.018. |
[10] |
C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem,, Master Thesis, (2011). Google Scholar |
[11] |
J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field,, Comm. Partial Differential Equations, 37 (2012), 1462.
doi: 10.1080/03605302.2012.679990. |
[12] |
J. Escher, P. Laurencot and C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS,, Trans. Amer. Math. Soc., ().
doi: 10.1090/S0002-9947-2014-06320-4. |
[13] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, vol. 20 of Courant Lect. Notes Math.,, Courant Inst. Math. Sci., (2010).
|
[14] |
M. Fila, B. Kawohl and H. A. Levine, Quenching for quasilinear equations,, Comm. Partial Differential Equations, 17 (1992), 593.
doi: 10.1080/03605309208820855. |
[15] |
P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem,, Commun. Contemp. Math., 9 (2007), 701.
doi: 10.1142/S0219199707002617. |
[16] |
P. Korman and Y. Li, Global solution curves for a class of quasilinear boundary-value problems,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1197.
doi: 10.1017/S0308210509001449. |
[17] |
T. Laetsch, The number of solutions of a nonlinear two point boundary value problem,, Indiana Univ. Math. J., 20 (1970), 1.
doi: 10.1512/iumj.1971.20.20001. |
[18] |
H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations,, Ann. Mat. Pura Appl. (4), 155 (1989), 243.
doi: 10.1007/BF01765943. |
[19] |
W. Li and Z. Liu, Exact number of solutions of a prescribed mean curvature equation,, J. Math. Anal. Appl., 367 (2010), 486.
doi: 10.1016/j.jmaa.2010.01.055. |
[20] |
A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics,, Methods Appl. Anal., 15 (2008), 297.
doi: 10.4310/MAA.2008.v15.n3.a4. |
[21] |
F. Obersnel, Classical and non-classical sign-changing solutions of a one-dimensional autonomous prescribed curvature equation,, Adv. Nonlinear Stud., 7 (2007), 671.
|
[22] |
H. Pan, One-dimensional prescribed mean curvature equation with exponential nonlinearity,, Nonlinear Anal., 70 (2009), 999.
doi: 10.1016/j.na.2008.01.027. |
[23] |
H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations,, Nonlinear Anal., 74 (2011), 1234.
doi: 10.1016/j.na.2010.09.063. |
[24] |
H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II,, Nonlinear Anal., 74 (2011), 3751.
doi: 10.1016/j.na.2011.03.020. |
[25] |
H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models,, Nonlinear Anal. Real World Appl., 13 (2012), 2432.
doi: 10.1016/j.nonrwa.2012.02.012. |
[26] |
H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions,, J. Differential Equations, 254 (2013), 1464.
doi: 10.1016/j.jde.2012.10.025. |
[27] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman & Hall/CRC, (2003).
|
[28] |
J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models,, J. Engrg. Math., 53 (2005), 239.
doi: 10.1007/s10665-005-9013-2. |
[29] |
J. Wei and D. Ye, On MEMS equation with fringing field,, Proc. Amer. Math. Soc., 138 (2010), 1693.
doi: 10.1090/S0002-9939-09-10226-5. |
[30] |
X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities,, J. Math. Anal. Appl., 395 (2013), 393.
doi: 10.1016/j.jmaa.2012.05.053. |
show all references
References:
[1] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities,, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63.
|
[2] |
D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation,, J. Differential Equations, 243 (2007), 208.
doi: 10.1016/j.jde.2007.05.031. |
[3] |
N. D. Brubaker and A. E. Lindsay, Analysis of the singular solution branch of a prescribed mean curvature equation with singular nonlinearity modeling a MEMS capacitor,, Preprint, (). Google Scholar |
[4] |
N. D. Brubaker and A. E. Lindsay, The onset of multivalued solutions of a prescribed mean curvature equation with singular nonlinearity,, Eur. J. Appl. Math., 24 (2013), 631.
doi: 10.1017/S0956792513000077. |
[5] |
N. D. Brubaker and J. A. Pelesko, Non-linear effects on canonical MEMS models,, Eur. J. Appl. Math., 22 (2011), 455.
doi: 10.1017/S0956792511000180. |
[6] |
N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity,, Nonlinear Anal., 75 (2012), 5086.
doi: 10.1016/j.na.2012.04.025. |
[7] |
N. D. Brubaker, J. I. Siddique, E. Sabo, R. Deaton and J. A. Pelesko, Refinements to the study of electrostatic deflections: theory and experiment,, Eur. J. Appl. Math., 24 (2013), 343.
doi: 10.1017/S0956792512000435. |
[8] |
M. Burns and M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux,, Eur. J. Appl. Math., 22 (2011), 317.
doi: 10.1017/S0956792511000076. |
[9] |
Y.-H. Cheng, K.-C. Hung and S.-H. Wang, Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS,, Nonlinear Anal., 89 (2013), 284.
doi: 10.1016/j.na.2013.04.018. |
[10] |
C.-H. Chuang, On Exact Multiplicity and Bifurcation Diagrams of Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem,, Master Thesis, (2011). Google Scholar |
[11] |
J. Dávila and J. Wei, Point ruptures for a MEMS equation with fringing field,, Comm. Partial Differential Equations, 37 (2012), 1462.
doi: 10.1080/03605302.2012.679990. |
[12] |
J. Escher, P. Laurencot and C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS,, Trans. Amer. Math. Soc., ().
doi: 10.1090/S0002-9947-2014-06320-4. |
[13] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, vol. 20 of Courant Lect. Notes Math.,, Courant Inst. Math. Sci., (2010).
|
[14] |
M. Fila, B. Kawohl and H. A. Levine, Quenching for quasilinear equations,, Comm. Partial Differential Equations, 17 (1992), 593.
doi: 10.1080/03605309208820855. |
[15] |
P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem,, Commun. Contemp. Math., 9 (2007), 701.
doi: 10.1142/S0219199707002617. |
[16] |
P. Korman and Y. Li, Global solution curves for a class of quasilinear boundary-value problems,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1197.
doi: 10.1017/S0308210509001449. |
[17] |
T. Laetsch, The number of solutions of a nonlinear two point boundary value problem,, Indiana Univ. Math. J., 20 (1970), 1.
doi: 10.1512/iumj.1971.20.20001. |
[18] |
H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations,, Ann. Mat. Pura Appl. (4), 155 (1989), 243.
doi: 10.1007/BF01765943. |
[19] |
W. Li and Z. Liu, Exact number of solutions of a prescribed mean curvature equation,, J. Math. Anal. Appl., 367 (2010), 486.
doi: 10.1016/j.jmaa.2010.01.055. |
[20] |
A. E. Lindsay and M. J. Ward, Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. I. Fold point asymptotics,, Methods Appl. Anal., 15 (2008), 297.
doi: 10.4310/MAA.2008.v15.n3.a4. |
[21] |
F. Obersnel, Classical and non-classical sign-changing solutions of a one-dimensional autonomous prescribed curvature equation,, Adv. Nonlinear Stud., 7 (2007), 671.
|
[22] |
H. Pan, One-dimensional prescribed mean curvature equation with exponential nonlinearity,, Nonlinear Anal., 70 (2009), 999.
doi: 10.1016/j.na.2008.01.027. |
[23] |
H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations,, Nonlinear Anal., 74 (2011), 1234.
doi: 10.1016/j.na.2010.09.063. |
[24] |
H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II,, Nonlinear Anal., 74 (2011), 3751.
doi: 10.1016/j.na.2011.03.020. |
[25] |
H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models,, Nonlinear Anal. Real World Appl., 13 (2012), 2432.
doi: 10.1016/j.nonrwa.2012.02.012. |
[26] |
H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions,, J. Differential Equations, 254 (2013), 1464.
doi: 10.1016/j.jde.2012.10.025. |
[27] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman & Hall/CRC, (2003).
|
[28] |
J. A. Pelesko and T. A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models,, J. Engrg. Math., 53 (2005), 239.
doi: 10.1007/s10665-005-9013-2. |
[29] |
J. Wei and D. Ye, On MEMS equation with fringing field,, Proc. Amer. Math. Soc., 138 (2010), 1693.
doi: 10.1090/S0002-9939-09-10226-5. |
[30] |
X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities,, J. Math. Anal. Appl., 395 (2013), 393.
doi: 10.1016/j.jmaa.2012.05.053. |
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