• Previous Article
    Global stability of a predator-prey system with stage structure and mutual interference
  • DCDS-B Home
  • This Issue
  • Next Article
    Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent
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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory,, Applied Mathematical Sciences 83, (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. 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. 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. 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. doi: 10.1063/1.870138.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[24]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135. 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. 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. 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, (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. doi: 10.1017/S0308210500025609.

[29]

H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Review, 32 (1990), 262. 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. 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), (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. 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. 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.

[35]

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

[36]

J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations,, Manuscripta Mathematica, 96 (1998), 113. 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, 11 (1998), 227.

[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.

[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. 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. 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. 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. 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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory,, Applied Mathematical Sciences 83, (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. 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. 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. 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. doi: 10.1063/1.870138.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[24]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dynamical Systems, 9 (2010), 1135. 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. 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. 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, (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. doi: 10.1017/S0308210500025609.

[29]

H. A. Levine, The role of critical exponents in blowup theorems,, SIAM Review, 32 (1990), 262. 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. 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), (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. 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. 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.

[35]

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

[36]

J. Wei, Conditions for two-peaked solutions of singularly perturbed elliptic equations,, Manuscripta Mathematica, 96 (1998), 113. 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, 11 (1998), 227.

[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.

[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. 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. 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. doi: 10.1137/060648866.

[1]

Avner Friedman, Kang-Ling Liao. The role of the cytokines IL-27 and IL-35 in cancer. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1203-1217. doi: 10.3934/mbe.2015.12.1203

[2]

Francisco Pedroche, Regino Criado, Esther García, Miguel Romance, Victoria E. Sánchez. Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks & Heterogeneous Media, 2015, 10 (1) : 101-125. doi: 10.3934/nhm.2015.10.101

[3]

Harsh Pittie and Arun Ram. A Pieri-Chevalley formula in the K-theory of aG/B-bundle. Electronic Research Announcements, 1999, 5: 102-107.

[4]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20

[5]

Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903

[6]

Liu Rui. The explicit nonlinear wave solutions of the generalized $b$-equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1029-1047. doi: 10.3934/cpaa.2013.12.1029

[7]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637

[8]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995

[9]

Marco Cabral, Ricardo Rosa, Roger Temam. Existence and dimension of the attractor for the Bénard problem on channel-like domains. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 89-116. doi: 10.3934/dcds.2004.10.89

[10]

Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283

[11]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047

[12]

Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329

[13]

B. A. Wagner, Andrea L. Bertozzi, L. E. Howle. Positive feedback control of Rayleigh-Bénard convection. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619

[14]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[15]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013

[16]

O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449

[17]

Seema Nanda, Lisette dePillis, Ami Radunskaya. B cell chronic lymphocytic leukemia - A model with immune response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1053-1076. doi: 10.3934/dcdsb.2013.18.1053

[18]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591

[19]

Chengzhi Li, Jianquan Li, Zhien Ma. Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1107-1116. doi: 10.3934/dcdsb.2015.20.1107

[20]

Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]