Citation: |
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.doi: 10.1080/03605307908820113. |
[2] |
K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source, C. R. Acad. Sci. Paris, 351 (2013), 585-591.doi: 10.1016/j.crma.2013.07.027. |
[3] |
J. Burczak, T. Cieslak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228.doi: 10.1016/j.na.2012.04.038. |
[4] |
X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.doi: 10.1002/mma.2992. |
[5] |
T. Cieslak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426.doi: 10.1016/j.jmaa.2006.03.080. |
[6] |
T. Cieslak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. I. H. Poincaré AN, 27 (2010), 437-446.doi: 10.1016/j.anihpc.2009.11.016. |
[7] |
T. Cieslak and C. Stinner, Finite-time blow up and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.doi: 10.1016/j.jde.2012.01.045. |
[8] |
T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.doi: 10.1007/s10440-013-9832-5. |
[9] |
T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.doi: 10.1088/0951-7715/21/5/009. |
[10] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[11] |
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, in Harmonic Analysis and Nonlinear Partial Differential Equations, Res. Inst. Math. Sci. (RIMS), Kyoto, 26 (2011), 159-175. |
[12] |
D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[13] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. |
[14] |
T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.doi: 10.1006/aama.2001.0721. |
[15] |
T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[16] |
T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.doi: 10.1007/s00285-008-0201-3. |
[17] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.doi: 10.1016/j.jde.2004.10.022. |
[18] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.doi: 10.1017/S0956792501004363. |
[19] |
S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Math. Methods Appl. Sci., 36 (2013), 745-760.doi: 10.1002/mma.2622. |
[20] |
S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.doi: 10.1016/j.jde.2014.01.028. |
[21] |
W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.doi: 10.1090/S0002-9947-1992-1046835-6. |
[22] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.doi: 10.1016/0022-5193(70)90092-5. |
[23] |
N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. I. H. Poincaré AN, 31 (2014), 851-875.doi: 10.1016/j.anihpc.2013.07.007. |
[24] |
C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal.-Real World Appl., 14 (2013), 1634-1642.doi: 10.1016/j.nonrwa.2012.10.022. |
[25] |
T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[26] |
E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.doi: 10.1016/j.na.2010.08.044. |
[27] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoret. Biol., 42 (1973), 63-105.doi: 10.1016/0022-5193(73)90149-5. |
[28] |
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737. |
[29] |
T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains, Funkcialaj Ekvacioj, 48 (2005), 247-271.doi: 10.1619/fesi.48.247. |
[30] |
T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. |
[31] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019. |
[32] |
Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.doi: 10.1016/j.jde.2011.07.010. |
[33] |
Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.doi: 10.1142/S0218202512500443. |
[34] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.doi: 10.1080/03605300701319003. |
[35] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Appl. Math. Sci., 68, Springer-Verlag, NewYork, 1997. |
[36] |
L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete and Continuous Dynamical Systems, 34 (2014), 789-802.doi: 10.3934/dcds.2014.34.789. |
[37] |
L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.doi: 10.1016/j.jde.2013.12.007. |
[38] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.doi: 10.1063/1.2766864. |
[39] |
Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190.doi: 10.1051/mmnp/20105311. |
[40] |
Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.doi: 10.1088/0951-7715/24/12/001. |
[41] |
Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.doi: 10.1137/110853972. |
[42] |
M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.doi: 10.1080/03605300903473426. |
[43] |
M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.doi: 10.1016/j.jmaa.2008.07.071. |
[44] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.doi: 10.1016/j.matpur.2013.01.020. |
[45] |
M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.doi: 10.1002/mma.1146. |
[46] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.doi: 10.1016/j.jde.2010.02.008. |
[47] |
M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.doi: 10.1016/j.na.2009.07.045. |
[48] |
D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding, Nonlinear Anal., 59 (2004), 1293-1310.doi: 10.1016/j.na.2004.08.015. |