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Blowup in higher dimensional two species chemotactic systems
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 |
References:
[1] |
P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Colloq. Math., 68 (1995), 229-239. |
[2] |
P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205. |
[3] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205. |
[3] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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