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Morrey spaces norms and criteria for blowup in chemotaxis models
1. | Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław |
2. | Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland |
References:
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P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205. |
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P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239. |
[3] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[4] |
P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.
doi: 10.1007/s00028-009-0048-0. |
[5] |
P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, arXiv:1410.7807 v.2. |
[6] |
P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems, Nonlinearity, 28 (2015), 4369-4387.
doi: 10.1088/0951-7715/28/12/4369. |
[7] |
P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126.
doi: 10.1002/mma.1036. |
[8] |
P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci., Mathematics, 63 (2015), 41-51.
doi: 10.4064/ba63-1-6. |
[9] |
G. Karch and K. Suzuki, Blow-up versus global existence of solutions to aggregation equations, Appl. Math. (Warsaw), 38 (2011), 243-258.
doi: 10.4064/am38-3-1. |
[10] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differ. Integral Equ., 16 (2003), 427-452. |
[11] |
P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Diff. Equ., 18 (2013), 1189-1208. |
[12] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
show all references
References:
[1] |
P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205. |
[2] |
P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III, Coll. Math., 68 (1995), 229-239. |
[3] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[4] |
P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.
doi: 10.1007/s00028-009-0048-0. |
[5] |
P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, arXiv:1410.7807 v.2. |
[6] |
P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems, Nonlinearity, 28 (2015), 4369-4387.
doi: 10.1088/0951-7715/28/12/4369. |
[7] |
P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126.
doi: 10.1002/mma.1036. |
[8] |
P. Biler and J. Zienkiewicz, Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci., Mathematics, 63 (2015), 41-51.
doi: 10.4064/ba63-1-6. |
[9] |
G. Karch and K. Suzuki, Blow-up versus global existence of solutions to aggregation equations, Appl. Math. (Warsaw), 38 (2011), 243-258.
doi: 10.4064/am38-3-1. |
[10] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differ. Integral Equ., 16 (2003), 427-452. |
[11] |
P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Diff. Equ., 18 (2013), 1189-1208. |
[12] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
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