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January  2014, 19(1): 189-215. doi: 10.3934/dcdsb.2014.19.189

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

Received  February 2013 Revised  October 2013 Published  December 2013

Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we predict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two spatial dimensions, the singular perturbation theory reveals that the set of possible singularity points depends subtly on the geometry of the domain and the equation parameters. In three spatial dimensions, preliminary numerical simulations indicate that the multiplicity of singularities can be even more complex. For the aforementioned scenarios, the analysis highlights the dichotomy of behaviors exhibited between the second and fourth order cases.
Citation: Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189
References:
[1]

C. Bandle and H. Brunner, Blowup in diffusion equations: A survey,, Journal of Computational and Applied Mathematics, 97 (1998), 3.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1989-0937878-9.  Google Scholar

[3]

V. A. Galaktionov and J.-L. Vázquez, The problem of blow-up in nonlinear parabolic equations,, DCDS-A, 8 (2002), 399.   Google Scholar

[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.  doi: 10.1137/S003613990241552X.  Google Scholar

[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.  doi: 10.1007/s10665-009-9343-6.  Google Scholar

[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.  doi: 10.1137/050643167.  Google Scholar

[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.  doi: 10.1137/110832550.  Google Scholar

[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.  doi: 10.1007/s00332-013-9169-2.  Google Scholar

[9]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana U. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[10]

J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory,, Applied Mathematical Sciences 83, (1989).   Google Scholar

[11]

A. J. Bernoff and T. P. Witelski, Stability and dynamics of self-similarity in evolution equations,, Journal of Engineering Mathematics, 66 (2010), 11.  doi: 10.1007/s10665-009-9309-8.  Google Scholar

[12]

A. J. Bernoff and T. P. Witelski, Dynamics of three-dimensional thin film rupture,, Physica D, 147 (2000), 155.  doi: 10.1016/S0167-2789(00)00165-2.  Google Scholar

[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.  doi: 10.1023/B:JOSS.0000033251.81126.af.  Google Scholar

[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.  doi: 10.1063/1.870138.  Google Scholar

[15]

A. L. Bertozzi, G. Grun and T. P. Witelski, Dewetting films: Bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569.  doi: 10.1088/0951-7715/14/6/309.  Google Scholar

[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.  doi: 10.1017/S0956792503005321.  Google Scholar

[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.  doi: 10.1088/0951-7715/22/7/012.  Google Scholar

[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.  doi: 10.1512/iumj.2002.51.2131.  Google Scholar

[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.   Google Scholar

[20]

G. Fibich, Self-focusing: Past and present, Topics in Applied Physics, 114 (2009), 413.   Google Scholar

[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.  doi: 10.1007/BF00383674.  Google Scholar

[22]

V. A. Galaktionov and M. Chaves, Regional Blow-up for a higher-order semilinear parabolic equations,, Euro. Jnl of Applied Mathematics, (2001), 601.  doi: 10.1017/S0956792501004685.  Google Scholar

[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.   Google Scholar

[24]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135.  doi: 10.1137/09077117X.  Google Scholar

[25]

Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior,, J. Diff. Eqns., 244 (2008), 2277.  doi: 10.1016/j.jde.2008.02.005.  Google Scholar

[26]

F. Gazzola and H.-C. Grunau, Some new properties of biharmonic heat kernels,, Nonlinear Analysis, 70 (2009), 2965.  doi: 10.1016/j.na.2008.12.039.  Google Scholar

[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, (2013).  doi: 10.1090/conm/594/11775.  Google Scholar

[28]

A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Royal Soc. Edinburgh Sect. A, 98 (1984), 203.  doi: 10.1017/S0308210500025609.  Google Scholar

[29]

H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Review, 32 (1990), 262.  doi: 10.1137/1032046.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6663.  Google Scholar

[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), (1985).  doi: 10.1007/978-1-4613-2349-5.  Google Scholar

[32]

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

[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.  doi: 10.1016/0022-0396(88)90138-6.  Google Scholar

[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.   Google Scholar

[35]

H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations,, Proc. Symp. Pure Math. Part I, 18 (1968), 138.   Google Scholar

[36]

J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations,, Manuscripta Mathematica, 96 (1998), 113.  doi: 10.1007/s002290050057.  Google Scholar

[37]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem,, Topological Methods in Nonlinear Analysis, 11 (1998), 227.   Google Scholar

[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.  doi: 10.1080/03605300008821511.  Google Scholar

[39]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2002).   Google Scholar

[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.  doi: 10.1137/040613391.  Google Scholar

[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.  doi: 10.1002/cpa.3160450703.  Google Scholar

[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.  doi: 10.1137/060648866.  Google Scholar

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.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1989-0937878-9.  Google Scholar

[3]

V. A. Galaktionov and J.-L. Vázquez, The problem of blow-up in nonlinear parabolic equations,, DCDS-A, 8 (2002), 399.   Google Scholar

[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.  doi: 10.1137/S003613990241552X.  Google Scholar

[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.  doi: 10.1007/s10665-009-9343-6.  Google Scholar

[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.  doi: 10.1137/050643167.  Google Scholar

[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.  doi: 10.1137/110832550.  Google Scholar

[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.  doi: 10.1007/s00332-013-9169-2.  Google Scholar

[9]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana U. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[10]

J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory,, Applied Mathematical Sciences 83, (1989).   Google Scholar

[11]

A. J. Bernoff and T. P. Witelski, Stability and dynamics of self-similarity in evolution equations,, Journal of Engineering Mathematics, 66 (2010), 11.  doi: 10.1007/s10665-009-9309-8.  Google Scholar

[12]

A. J. Bernoff and T. P. Witelski, Dynamics of three-dimensional thin film rupture,, Physica D, 147 (2000), 155.  doi: 10.1016/S0167-2789(00)00165-2.  Google Scholar

[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.  doi: 10.1023/B:JOSS.0000033251.81126.af.  Google Scholar

[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.  doi: 10.1063/1.870138.  Google Scholar

[15]

A. L. Bertozzi, G. Grun and T. P. Witelski, Dewetting films: Bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569.  doi: 10.1088/0951-7715/14/6/309.  Google Scholar

[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.  doi: 10.1017/S0956792503005321.  Google Scholar

[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.  doi: 10.1088/0951-7715/22/7/012.  Google Scholar

[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.  doi: 10.1512/iumj.2002.51.2131.  Google Scholar

[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.   Google Scholar

[20]

G. Fibich, Self-focusing: Past and present, Topics in Applied Physics, 114 (2009), 413.   Google Scholar

[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.  doi: 10.1007/BF00383674.  Google Scholar

[22]

V. A. Galaktionov and M. Chaves, Regional Blow-up for a higher-order semilinear parabolic equations,, Euro. Jnl of Applied Mathematics, (2001), 601.  doi: 10.1017/S0956792501004685.  Google Scholar

[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.   Google Scholar

[24]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135.  doi: 10.1137/09077117X.  Google Scholar

[25]

Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior,, J. Diff. Eqns., 244 (2008), 2277.  doi: 10.1016/j.jde.2008.02.005.  Google Scholar

[26]

F. Gazzola and H.-C. Grunau, Some new properties of biharmonic heat kernels,, Nonlinear Analysis, 70 (2009), 2965.  doi: 10.1016/j.na.2008.12.039.  Google Scholar

[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, (2013).  doi: 10.1090/conm/594/11775.  Google Scholar

[28]

A. A. Lacey, The form of blow-up for nonlinear parabolic equations,, Proc. Royal Soc. Edinburgh Sect. A, 98 (1984), 203.  doi: 10.1017/S0308210500025609.  Google Scholar

[29]

H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Review, 32 (1990), 262.  doi: 10.1137/1032046.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6663.  Google Scholar

[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), (1985).  doi: 10.1007/978-1-4613-2349-5.  Google Scholar

[32]

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

[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.  doi: 10.1016/0022-0396(88)90138-6.  Google Scholar

[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.   Google Scholar

[35]

H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations,, Proc. Symp. Pure Math. Part I, 18 (1968), 138.   Google Scholar

[36]

J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations,, Manuscripta Mathematica, 96 (1998), 113.  doi: 10.1007/s002290050057.  Google Scholar

[37]

E. N. Dancer and J. Wei, On the effect of domain topology in a singular perturbation problem,, Topological Methods in Nonlinear Analysis, 11 (1998), 227.   Google Scholar

[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.  doi: 10.1080/03605300008821511.  Google Scholar

[39]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2002).   Google Scholar

[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.  doi: 10.1137/040613391.  Google Scholar

[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.  doi: 10.1002/cpa.3160450703.  Google Scholar

[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.  doi: 10.1137/060648866.  Google Scholar

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