American Institute of Mathematical Sciences

Advanced
November  2013, 12(6): 2627-2644. doi: 10.3934/cpaa.2013.12.2627

Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two

Elio E. Espejo 1, , Masaki Kurokiba 2, and  Takashi Suzuki 3,

1. 

Departamento de Matemáticas, Universidad de los Andes, Bogotá
2. 

Muroran Institute of Technology, 27-1 Mizumoto, Muroran, 050-8585, Japan
3. 

Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531

Received  August 2012 Revised  January 2013 Published  May 2013

We study a drift-diffusion system on bounded domain in two-space dimension. This model is provided with a hetero-separative and homo-aggregative feature subject to a gradient of physical or chemical potential which is proportional to their densities. We extend a criterion of global-in-time existence of the solution, especially for non-radially symmetric case. Then we perform the blowup analysis such as the formation of collapses and collapse mass separations. A slightly different model describing cross chemotaxis is also discussed.
Keywords: formation of collapse, blowup threshold, mass separation., Drift-diffusion model, chemotaxis.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92C15, 92C1.
Citation: Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 0.1080/03605307908820113.  Google Scholar

[2]

P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.  Google Scholar

[3]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis, 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[4]

P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitating interaction of particles, II, Colloq. Math., 67 (1994), 297-308.  Google Scholar

[5]

C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Math Letters, (2012), 352-356. doi: 10.1016/j.aml.2011.09.013.  Google Scholar

[6]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete and Continuous Dynamical Systems B, 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.  Google Scholar

[7]

E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential and Integral Equations, 25 (2012), 251-288.  Google Scholar

[8]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.  Google Scholar

[9]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential and Integral Equations, 23 (2010), 451-462.  Google Scholar

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar

[12]

T. Iwaniec and A. Verde, On the operator $L(f) = f \log |f|$, J. Funct. Anal., 169 (1999), 391-420. doi: 10.1006/jfan.1999.3443.  Google Scholar

[13]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  Google Scholar

[14]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential and Integral Equations, 4 (2003), 427-452.  Google Scholar

[15]

M. Kurokiba and T. Ogawa, Wellposedness of the drit-diffusion system in $L^p$ arising from the semiconductor device simulation, J. Math. Anal. Appl., 342 (2008), 1052-1067. doi: 10.1016/j.jmaa.2007.11.017.  Google Scholar

[16]

M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Comm. Pure Appl. Anal., 5 (2006), 97-106. doi: 10.3934/cpaa.2006.5.97.  Google Scholar

[17]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[18]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.  Google Scholar

[19]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  Google Scholar

[20]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562.  Google Scholar

[21]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.  Google Scholar

[22]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.  Google Scholar

[23]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-368.  Google Scholar

[24]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51. doi: 10.1006/jfan.2001.3802.  Google Scholar

[25]

I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems, J. Euro. Math. Soc., 7 (2005), 413-448. doi: 10.4171/JEMS/34.  Google Scholar

[26]

T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005  Google Scholar

[27]

T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008.  Google Scholar

[28]

T. Suzuki, 2D Brownian point vortices and the drift-diffusion model,, Discrete and Continuous Dynamical Systems Ser. S., ().   Google Scholar

[29]

T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part,, J. Math. Pure Appl., ().  doi: 10.1016/j.matpur.2013.01.004.  Google Scholar

[30]

T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011.  Google Scholar

[31]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661. doi: 10.1017/S0956792501004843.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 0.1080/03605307908820113.  Google Scholar

[2]

P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.  Google Scholar

[3]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis, 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[4]

P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitating interaction of particles, II, Colloq. Math., 67 (1994), 297-308.  Google Scholar

[5]

C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Math Letters, (2012), 352-356. doi: 10.1016/j.aml.2011.09.013.  Google Scholar

[6]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete and Continuous Dynamical Systems B, 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.  Google Scholar

[7]

E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential and Integral Equations, 25 (2012), 251-288.  Google Scholar

[8]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338. doi: 10.1524/anly.2009.1029.  Google Scholar

[9]

E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential and Integral Equations, 23 (2010), 451-462.  Google Scholar

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar

[12]

T. Iwaniec and A. Verde, On the operator $L(f) = f \log |f|$, J. Funct. Anal., 169 (1999), 391-420. doi: 10.1006/jfan.1999.3443.  Google Scholar

[13]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  Google Scholar

[14]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential and Integral Equations, 4 (2003), 427-452.  Google Scholar

[15]

M. Kurokiba and T. Ogawa, Wellposedness of the drit-diffusion system in $L^p$ arising from the semiconductor device simulation, J. Math. Anal. Appl., 342 (2008), 1052-1067. doi: 10.1016/j.jmaa.2007.11.017.  Google Scholar

[16]

M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Comm. Pure Appl. Anal., 5 (2006), 97-106. doi: 10.3934/cpaa.2006.5.97.  Google Scholar

[17]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[18]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.  Google Scholar

[19]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  Google Scholar

[20]

F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562.  Google Scholar

[21]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.  Google Scholar

[22]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.  Google Scholar

[23]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-368.  Google Scholar

[24]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51. doi: 10.1006/jfan.2001.3802.  Google Scholar

[25]

I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems, J. Euro. Math. Soc., 7 (2005), 413-448. doi: 10.4171/JEMS/34.  Google Scholar

[26]

T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005  Google Scholar

[27]

T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008.  Google Scholar

[28]

T. Suzuki, 2D Brownian point vortices and the drift-diffusion model,, Discrete and Continuous Dynamical Systems Ser. S., ().   Google Scholar

[29]

T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part,, J. Math. Pure Appl., ().  doi: 10.1016/j.matpur.2013.01.004.  Google Scholar

[30]

T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011.  Google Scholar

[31]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661. doi: 10.1017/S0956792501004843.  Google Scholar

[1]

H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 319-334. doi: 10.3934/dcdsb.2005.5.319
[2]

Dietmar Oelz, Alex Mogilner. A drift-diffusion model for molecular motor transport in anisotropic filament bundles. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4553-4567. doi: 10.3934/dcds.2016.36.4553
[3]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97
[4]

Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029
[5]

Clément Jourdana, Paola Pietra. A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures. Kinetic & Related Models, 2019, 12 (1) : 217-242. doi: 10.3934/krm.2019010
[6]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449
[7]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[8]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558
[9]

Claire Chainais-Hillairet, Ingrid Lacroix-Violet. On the existence of solutions for a drift-diffusion system arising in corrosion modeling. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 77-92. doi: 10.3934/dcdsb.2015.20.77
[10]

Kelei Wang. The singular limit problem in a phase separation model with different diffusion rates $^*$. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 483-512. doi: 10.3934/dcds.2015.35.483
[11]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[12]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191
[13]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4867-4885. doi: 10.3934/dcdsb.2020316
[14]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[15]

Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021237
[16]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[17]

Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823
[18]

Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks & Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239
[19]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157
[20]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy