# American Institute of Mathematical Sciences

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

 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.
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

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##### 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
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