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January  2013, 12(1): 89-98. doi: 10.3934/cpaa.2013.12.89

Blowup in higher dimensional two species chemotactic systems

Piotr Biler 1, , Elio E. Espejo 2, and  Ignacio Guerra 3,

1. 

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław, Poland
2. 

Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia
3. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile

Received  July 2011 Revised  February 2012 Published  September 2012

This paper deals with blowup properties of solutions to multicomponent parabolic-elliptic Keller--Segel model of chemotaxis in higher dimensions.
Keywords: multicomponent Keller--Segel model, blowup of solutions., Chemotaxis.
Mathematics Subject Classification: Primary: 35Q92, 35.
Citation: Piotr Biler, Elio E. Espejo, Ignacio Guerra. Blowup in higher dimensional two species chemotactic systems. Communications on Pure & Applied Analysis, 2013, 12 (1) : 89-98. doi: 10.3934/cpaa.2013.12.89
References:
[1]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Colloq. Math., 68 (1995), 229-239.  Google Scholar

[2]

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.  Google Scholar

[3]

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

[4]

P. Biler, M. Cannone, I. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708. doi: DOI: 10.1007/s00208-004-0565-7.  Google Scholar

[5]

P. Biler and J. Dolbeault, Long time behaviour of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472. doi: DOI: 1424-0637/00/030461-12.  Google Scholar

[6]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis TMA, 23 (1994), 1189-1209.  Google Scholar

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Eq., 10 (2010), 247-262. doi: DOI 10.1007/s00028-009-0048-0.  Google Scholar

[8]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 1-32.  Google Scholar

[9]

V. Calvez, L. Corrias and M. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Partial Diff. Eq., 37 (2012), 561-584. doi: DOI: 10.1080/03605302.2012.655824.  Google Scholar

[10]

C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion System, Applied Mathematics Letters, 25 (2012), 352-356. doi: doi:10.1016/j.aml.2011.09.013.  Google Scholar

[11]

C. Conca, E. E. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two-species chemotactic Keller-Segel system in $R^2$, Euro. J. Appl. Math, 22 (2011), 553-580. doi: doi:10.1017/S0956792511000258.  Google Scholar

[12]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comp. Modelling, 47 (2008), 755-764. doi: doi:10.1016/j.mcm.2007.06.005.  Google Scholar

[13]

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

[14]

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

[15]

H. Kozono and Y. Sugiyama, The Keller-Segel model of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ spaces with applications to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500.  Google Scholar

[16]

H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Eq., 247 (2009), 1-32. doi: doi:10.1016/j.jde.2009.03.027.  Google Scholar

[17]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations, 16 (2003), 427-452.  Google Scholar

[18]

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

[19]

A. Raczyński, Weak-$L^p$ solutions for a model of self-gravitating particles with an external potential, Studia Math., 179 (2007), 199-216. doi: doi:10.4064/sm179-3-1.  Google Scholar

[20]

A. Raczyński, Stability property of the two-dimensional Keller-Segel model, Asymptotic Analysis, 61 (2009), 35-59. doi: DOI 10.3233/ASY-2008-0907.  Google Scholar

[21]

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

[22]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: DOI 10.1017/S0956792501004843.  Google Scholar

show all references

References:
[1]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Colloq. Math., 68 (1995), 229-239.  Google Scholar

[2]

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.  Google Scholar

[3]

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

[4]

P. Biler, M. Cannone, I. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708. doi: DOI: 10.1007/s00208-004-0565-7.  Google Scholar

[5]

P. Biler and J. Dolbeault, Long time behaviour of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré, 1 (2000), 461-472. doi: DOI: 1424-0637/00/030461-12.  Google Scholar

[6]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis TMA, 23 (1994), 1189-1209.  Google Scholar

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Eq., 10 (2010), 247-262. doi: DOI 10.1007/s00028-009-0048-0.  Google Scholar

[8]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 1-32.  Google Scholar

[9]

V. Calvez, L. Corrias and M. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Partial Diff. Eq., 37 (2012), 561-584. doi: DOI: 10.1080/03605302.2012.655824.  Google Scholar

[10]

C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion System, Applied Mathematics Letters, 25 (2012), 352-356. doi: doi:10.1016/j.aml.2011.09.013.  Google Scholar

[11]

C. Conca, E. E. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two-species chemotactic Keller-Segel system in $R^2$, Euro. J. Appl. Math, 22 (2011), 553-580. doi: doi:10.1017/S0956792511000258.  Google Scholar

[12]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comp. Modelling, 47 (2008), 755-764. doi: doi:10.1016/j.mcm.2007.06.005.  Google Scholar

[13]

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

[14]

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

[15]

H. Kozono and Y. Sugiyama, The Keller-Segel model of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ spaces with applications to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500.  Google Scholar

[16]

H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Eq., 247 (2009), 1-32. doi: doi:10.1016/j.jde.2009.03.027.  Google Scholar

[17]

M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations, 16 (2003), 427-452.  Google Scholar

[18]

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

[19]

A. Raczyński, Weak-$L^p$ solutions for a model of self-gravitating particles with an external potential, Studia Math., 179 (2007), 199-216. doi: doi:10.4064/sm179-3-1.  Google Scholar

[20]

A. Raczyński, Stability property of the two-dimensional Keller-Segel model, Asymptotic Analysis, 61 (2009), 35-59. doi: DOI 10.3233/ASY-2008-0907.  Google Scholar

[21]

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

[22]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: DOI 10.1017/S0956792501004843.  Google Scholar

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