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An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations
1. | Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, United States |
References:
[1] |
C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, Journal of Computational and Applied Mathematics, 97 (1998), 3-22.
doi: 10.1016/S0377-0427(98)00100-9. |
[2] |
C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Transactions of the American Mathematical Society, 316 (1989), 595-622.
doi: 10.1090/S0002-9947-1989-0937878-9. |
[3] |
V. A. Galaktionov and J.-L. Vázquez, The problem of blow-up in nonlinear parabolic equations, DCDS-A, 8 (2002), 399-433. |
[4] |
C. J. Budd, V. A. Galaktionov and J. F. Williams, Self-similar blow-up in higher-order semilinear parabolic equations, SIAM J. Appl. Math, 64 (2004), 1775-1809.
doi: 10.1137/S003613990241552X. |
[5] |
C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry, J. Engineering Mathematics, 66 (2010), 217-236.
doi: 10.1007/s10665-009-9343-6. |
[6] |
R. D. Russell, J. F. Williams and X. Xu, MOVCOL4: A moving mesh code for fourth-order time-dependent partial differential equations, SIAM J. Sci. Comput., 29 (2007), 197-220 (electronic).
doi: 10.1137/050643167. |
[7] |
A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor, SIAM J. Appl. Math., 72 (2012), 935-958.
doi: 10.1137/110832550. |
[8] |
A. E. Lindsay, J. Lega and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries, Journal of Nonlinear Science, 23 (2013), 807-834.
doi: 10.1007/s00332-013-9169-2. |
[9] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana U. Math. J., 34 (1985), 425-447.
doi: 10.1512/iumj.1985.34.34025. |
[10] |
J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences 83, Springer-Verlag, 1989. |
[11] |
A. J. Bernoff and T. P. Witelski, Stability and dynamics of self-similarity in evolution equations, Journal of Engineering Mathematics, 66 (2010), 11-31.
doi: 10.1007/s10665-009-9309-8. |
[12] |
A. J. Bernoff and T. P. Witelski, Dynamics of three-dimensional thin film rupture, Physica D, 147 (2000), 155-176.
doi: 10.1016/S0167-2789(00)00165-2. |
[13] |
A. J. Bernoff, A. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Stat. Phys., 93 (1998), 725-776.
doi: 10.1023/B:JOSS.0000033251.81126.af. |
[14] |
A. J. Bernoff and T. P. Witelski, Stability of self-similar solutions for van der Waals driven thin film rupture, Physics of Fluids, 11 (1999), 2443-2445.
doi: 10.1063/1.870138. |
[15] |
A. L. Bertozzi, G. Grun and T. P. Witelski, Dewetting films: Bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592.
doi: 10.1088/0951-7715/14/6/309. |
[16] |
V. A. Galaktionov and J. F. Williams, Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory, Euro. Jnl Applied Mathematics, 14 (2003), 745-764.
doi: 10.1017/S0956792503005321. |
[17] |
V. A. Galaktionov, Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytical-numerical approach, Nonlinearity, 22 (2009), 1695-1741.
doi: 10.1088/0951-7715/22/7/012. |
[18] |
V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338.
doi: 10.1512/iumj.2002.51.2131. |
[19] |
D. A. Frank-Kamenetskii, Towards temperature distributions in a reaction vessel and the stationary theory of thermal explosion, Dokl. Acad. Nauk SSSR, 18 (1938), 411-412. |
[20] |
G. Fibich, Self-focusing: Past and present Topics in Applied Physics, 114 (2009), 413-438. |
[21] |
V. A. Galaktionov and J. L. Velázquez, Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225-244.
doi: 10.1007/BF00383674. |
[22] |
V. A. Galaktionov and M. Chaves, Regional Blow-up for a higher-order semilinear parabolic equations, Euro. Jnl of Applied Mathematics, (2001), 601-623.
doi: 10.1017/S0956792501004685. |
[23] |
J. J. L. Velázquez, V. A. Galaktionov, S. A. Posashkov and M. A. Herrero, On a general approach to extinction and blow-up for quasi-linear heat equations, Zh. Vychisl. Mat. i Mat. Fiz., 33 (1993), 246-258; Translation in Comput. Math. Math. Phys., 33 (1993), 217-227. |
[24] |
Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135-1163.
doi: 10.1137/09077117X. |
[25] |
Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, J. Diff. Eqns., 244 (2008), 2277-2309.
doi: 10.1016/j.jde.2008.02.005. |
[26] |
F. Gazzola and H.-C. Grunau, Some new properties of biharmonic heat kernels, Nonlinear Analysis, 70 (2009), 2965-2973.
doi: 10.1016/j.na.2008.12.039. |
[27] |
G. Barbatis and F. Gazzola, Higher order linear parabolic equations, Contemporary Mathematics series of the AMS: Recent Trends in Nonlinear Partial Differential Equations I: Evolution Problems, 594 (2013).
doi: 10.1090/conm/594/11775. |
[28] |
A. A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Royal Soc. Edinburgh Sect. A, 98 (1984), 203-214.
doi: 10.1017/S0308210500025609. |
[29] |
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.
doi: 10.1137/1032046. |
[30] |
K. Deng and H. A. Levine, The Role of Critical Exponents in Blow-Up Theorems: The Sequel, Journal of Mathematical Analysis and Applications, 243 (2000), 85-126.
doi: 10.1006/jmaa.1999.6663. |
[31] |
Y. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau (Plenum), New York, London, 1985.
doi: 10.1007/978-1-4613-2349-5. |
[32] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math, 16 (1963), 305-333.
doi: 10.1002/cpa.3160160307. |
[33] |
A. Friedman and L. Oswald, The blow-up time for higher order semilinear parabolic equations with small leading coefficients, Journal of Differential Equations, 75 (1988), 239-263.
doi: 10.1016/0022-0396(88)90138-6. |
[34] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokoyo Sect. IA Math., 13 (1966), 109-124. |
[35] |
H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math. Part I, Amer. Math. Soc., 18 (1968), 138-161. |
[36] |
J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations, Manuscripta Mathematica, 96 (1998), 113-131.
doi: 10.1007/s002290050057. |
[37] |
E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, 11 (1998), 227-248. |
[38] |
M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems, Communications in Partial Differential Equations, 25 (2000), 155-177.
doi: 10.1080/03605300008821511. |
[39] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002. |
[40] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338.
doi: 10.1137/040613391. |
[41] |
S. Filippas and R. V. Kohn, Refined asymptotics for the blowup of $u_t -\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[42] |
G. Flores, G. Mercado, J. A. Pelesko and N. & Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM Journal on Applied Mathematics, 67 (2007), 434-446.
doi: 10.1137/060648866. |
show all references
References:
[1] |
C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, Journal of Computational and Applied Mathematics, 97 (1998), 3-22.
doi: 10.1016/S0377-0427(98)00100-9. |
[2] |
C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Transactions of the American Mathematical Society, 316 (1989), 595-622.
doi: 10.1090/S0002-9947-1989-0937878-9. |
[3] |
V. A. Galaktionov and J.-L. Vázquez, The problem of blow-up in nonlinear parabolic equations, DCDS-A, 8 (2002), 399-433. |
[4] |
C. J. Budd, V. A. Galaktionov and J. F. Williams, Self-similar blow-up in higher-order semilinear parabolic equations, SIAM J. Appl. Math, 64 (2004), 1775-1809.
doi: 10.1137/S003613990241552X. |
[5] |
C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry, J. Engineering Mathematics, 66 (2010), 217-236.
doi: 10.1007/s10665-009-9343-6. |
[6] |
R. D. Russell, J. F. Williams and X. Xu, MOVCOL4: A moving mesh code for fourth-order time-dependent partial differential equations, SIAM J. Sci. Comput., 29 (2007), 197-220 (electronic).
doi: 10.1137/050643167. |
[7] |
A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor, SIAM J. Appl. Math., 72 (2012), 935-958.
doi: 10.1137/110832550. |
[8] |
A. E. Lindsay, J. Lega and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries, Journal of Nonlinear Science, 23 (2013), 807-834.
doi: 10.1007/s00332-013-9169-2. |
[9] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana U. Math. J., 34 (1985), 425-447.
doi: 10.1512/iumj.1985.34.34025. |
[10] |
J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences 83, Springer-Verlag, 1989. |
[11] |
A. J. Bernoff and T. P. Witelski, Stability and dynamics of self-similarity in evolution equations, Journal of Engineering Mathematics, 66 (2010), 11-31.
doi: 10.1007/s10665-009-9309-8. |
[12] |
A. J. Bernoff and T. P. Witelski, Dynamics of three-dimensional thin film rupture, Physica D, 147 (2000), 155-176.
doi: 10.1016/S0167-2789(00)00165-2. |
[13] |
A. J. Bernoff, A. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Stat. Phys., 93 (1998), 725-776.
doi: 10.1023/B:JOSS.0000033251.81126.af. |
[14] |
A. J. Bernoff and T. P. Witelski, Stability of self-similar solutions for van der Waals driven thin film rupture, Physics of Fluids, 11 (1999), 2443-2445.
doi: 10.1063/1.870138. |
[15] |
A. L. Bertozzi, G. Grun and T. P. Witelski, Dewetting films: Bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592.
doi: 10.1088/0951-7715/14/6/309. |
[16] |
V. A. Galaktionov and J. F. Williams, Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory, Euro. Jnl Applied Mathematics, 14 (2003), 745-764.
doi: 10.1017/S0956792503005321. |
[17] |
V. A. Galaktionov, Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytical-numerical approach, Nonlinearity, 22 (2009), 1695-1741.
doi: 10.1088/0951-7715/22/7/012. |
[18] |
V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338.
doi: 10.1512/iumj.2002.51.2131. |
[19] |
D. A. Frank-Kamenetskii, Towards temperature distributions in a reaction vessel and the stationary theory of thermal explosion, Dokl. Acad. Nauk SSSR, 18 (1938), 411-412. |
[20] |
G. Fibich, Self-focusing: Past and present Topics in Applied Physics, 114 (2009), 413-438. |
[21] |
V. A. Galaktionov and J. L. Velázquez, Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225-244.
doi: 10.1007/BF00383674. |
[22] |
V. A. Galaktionov and M. Chaves, Regional Blow-up for a higher-order semilinear parabolic equations, Euro. Jnl of Applied Mathematics, (2001), 601-623.
doi: 10.1017/S0956792501004685. |
[23] |
J. J. L. Velázquez, V. A. Galaktionov, S. A. Posashkov and M. A. Herrero, On a general approach to extinction and blow-up for quasi-linear heat equations, Zh. Vychisl. Mat. i Mat. Fiz., 33 (1993), 246-258; Translation in Comput. Math. Math. Phys., 33 (1993), 217-227. |
[24] |
Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135-1163.
doi: 10.1137/09077117X. |
[25] |
Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, J. Diff. Eqns., 244 (2008), 2277-2309.
doi: 10.1016/j.jde.2008.02.005. |
[26] |
F. Gazzola and H.-C. Grunau, Some new properties of biharmonic heat kernels, Nonlinear Analysis, 70 (2009), 2965-2973.
doi: 10.1016/j.na.2008.12.039. |
[27] |
G. Barbatis and F. Gazzola, Higher order linear parabolic equations, Contemporary Mathematics series of the AMS: Recent Trends in Nonlinear Partial Differential Equations I: Evolution Problems, 594 (2013).
doi: 10.1090/conm/594/11775. |
[28] |
A. A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Royal Soc. Edinburgh Sect. A, 98 (1984), 203-214.
doi: 10.1017/S0308210500025609. |
[29] |
H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.
doi: 10.1137/1032046. |
[30] |
K. Deng and H. A. Levine, The Role of Critical Exponents in Blow-Up Theorems: The Sequel, Journal of Mathematical Analysis and Applications, 243 (2000), 85-126.
doi: 10.1006/jmaa.1999.6663. |
[31] |
Y. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau (Plenum), New York, London, 1985.
doi: 10.1007/978-1-4613-2349-5. |
[32] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math, 16 (1963), 305-333.
doi: 10.1002/cpa.3160160307. |
[33] |
A. Friedman and L. Oswald, The blow-up time for higher order semilinear parabolic equations with small leading coefficients, Journal of Differential Equations, 75 (1988), 239-263.
doi: 10.1016/0022-0396(88)90138-6. |
[34] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokoyo Sect. IA Math., 13 (1966), 109-124. |
[35] |
H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math. Part I, Amer. Math. Soc., 18 (1968), 138-161. |
[36] |
J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations, Manuscripta Mathematica, 96 (1998), 113-131.
doi: 10.1007/s002290050057. |
[37] |
E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, 11 (1998), 227-248. |
[38] |
M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems, Communications in Partial Differential Equations, 25 (2000), 155-177.
doi: 10.1080/03605300008821511. |
[39] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002. |
[40] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338.
doi: 10.1137/040613391. |
[41] |
S. Filippas and R. V. Kohn, Refined asymptotics for the blowup of $u_t -\Delta u = u^p$, Comm. Pure Appl. Math., 45 (1992), 821-869.
doi: 10.1002/cpa.3160450703. |
[42] |
G. Flores, G. Mercado, J. A. Pelesko and N. & Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM Journal on Applied Mathematics, 67 (2007), 434-446.
doi: 10.1137/060648866. |
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