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

Previous Article
Decay rates for Kirchhoff-Timoshenko transmission problems
CPAA Home
This Issue
Next Article
A stability result for the Stokes-Boussinesq equations in infinite 3d channels

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

Abstract
Related Papers

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 and 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.
[2]	P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.
[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.
[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.
[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.
[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.
[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.
[17]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[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.
[19]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[20]	F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562.
[21]	M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.
[22]	T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.
[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.
[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.
[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.
[26]	T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005
[27]	T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008.
[28]	T. Suzuki, 2D Brownian point vortices and the drift-diffusion model, Discrete and Continuous Dynamical Systems Ser. S., accepted for publication.
[29]	T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part, J. Math. Pure Appl., accepted for publication. doi: 10.1016/j.matpur.2013.01.004.
[30]	T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011.
[31]	G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661. doi: 10.1017/S0956792501004843.

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.
[2]	P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.
[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.
[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.
[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.
[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.
[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.
[17]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[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.
[19]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[20]	F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562.
[21]	M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.
[22]	T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.
[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.
[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.
[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.
[26]	T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005
[27]	T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008.
[28]	T. Suzuki, 2D Brownian point vortices and the drift-diffusion model, Discrete and Continuous Dynamical Systems Ser. S., accepted for publication.
[29]	T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part, J. Math. Pure Appl., accepted for publication. doi: 10.1016/j.matpur.2013.01.004.
[30]	T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011.
[31]	G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661. doi: 10.1017/S0956792501004843.

[1]	H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2015, 20 (1) : 77-92. doi: 10.3934/dcdsb.2015.20.77
[10]	Luigi Barletti, Philipp Holzinger, Ansgar Jüngel. Formal derivation of quantum drift-diffusion equations with spin-orbit interaction. Kinetic and Related Models, 2022, 15 (2) : 257-282. doi: 10.3934/krm.2022007
[11]	Kelei Wang. The singular limit problem in a phase separation model with different diffusion rates $^$. Discrete and Continuous Dynamical Systems*, 2015, 35 (1) : 483-512. doi: 10.3934/dcds.2015.35.483
[12]	Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[13]	Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191
[14]	Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4867-4885. doi: 10.3934/dcdsb.2020316
[15]	Wenjing Wu, Tianli Jiang, Weiwei Liu, Jinliang Wang. Threshold dynamics of a reaction-diffusion cholera model with seasonality and nonlocal delay. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022099
[16]	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 and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[17]	Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4473-4489. doi: 10.3934/dcdsb.2021237
[18]	Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[19]	Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823
[20]	Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks and Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Elio E. Espejo Masaki Kurokiba Takashi Suzuki
on Google Scholar
Elio E. Espejo Masaki Kurokiba Takashi Suzuki

[Back to Top]