Stationary solutions for some shadow system of the KellerSegel model with logistic growth
Pages: 1023  1034,
Issue 5,
October
2015
doi:10.3934/dcdss.2015.8.1023 Abstract
References
Full text (627.7K)
Related Articles
Tohru Tsujikawa  Faculty of Engineering, University of Miyazaki, Miyazaki, 8892192, Japan (email)
Kousuke Kuto  Department of Communication Engineering and Informatics, The University of ElectroCommunications, Tokyo, 1828585, Japan (email)
Yasuhito Miyamoto  Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 1538914, Japan (email)
Hirofumi Izuhara  Faculty of Engineering, University of Miyazaki, Miyazaki, 8892192, Japan (email)
1 
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453474. 

2 
W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691722. 

3 
P. Biler, Local and global solvability of some parabolic system modelling chemotaxis, Adv. Mathe. Sci. and Appl., 8 (1998), 715743. 

4 
N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. Anal., 4 (1974), 1737. 

5 
E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO 2000, Continuation and bifurcation software for ordinary differential equations. 

6 
S.I. Ei, H. Izuhara and M. Mimura, Spatiotemporal oscillations in the KellerSegel system with logistic growth, Physica D, 277 (2014), 121. 

7 
C. Gai, Q. Wang and J. Yan, Qualitative analysis of stationary KellerSegel chemotaxis models with logistic growth, preprint, arXiv:1312.0258. 

8 
D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxisgrowth model, Sci. Math. Jpn, 70 (2009), 205211. 

9 
Y. Kabeya and W.M. Ni, Stationary KellerSegel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 4465. 

10 
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399415. 

11 
N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to MimuraTsujikawa model in one dimension, Math. Sci. Appl., 29 (2008), 265278. 

12 
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern in a chemotaxisdiffusiongrowth model, Physica D, 241 (2012), 16291639. 

13 
K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbateinduced phase transition model: II. Shadow system, Nonlinearity, 26 (2013), 13131343. 

14 
K. Kuto and T. Tsujikawa, Bifurcation structure of steadystates for bistable equations with nonlocal constraint, Discrete Continuous Dynam. Systems, Supplement volume, (2013), 455464. 

15 
K. Kuto and T. Tsujikawa, Limiting structure of steadystates to the LotkaVolterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 18011858. 

16 
K. Kuto and T. Tsujikawa, Bifurcation structure of steadystates for generalized AllenCahn equations with nonlocal constraint, preprint. 

17 
D. A. Lauffenburger and C. R. Kennedy, Localized bacterial infection in a chemotaxisdiffusiongrowth model, J. Math. Biol., 16 (1983), 141163. 

18 
C.S. Lin, W.N. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 127. 

19 
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern formation, Bull. Math. Biol., 53 (1991), 701719. 

20 
M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499543. 

21 
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Analysis, Theory, Methods and Applications, 51 (2002), 119144. 

22 
K. J. Painter and T. Hillen, Spatiotemporal chaos in a chemotaxis model, Physica D, 240 (2011), 363375. 

23 
R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math., 364 (1984), 131. 

24 
R. Schaaf, Global Solution Branches of TwoPoint Boundary Value Problems, Lecture Notes in Mathematics, 1458, SpringerVerlag, Berlin, 1990. 

25 
T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191224. 

26 
J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc., 354 (2002), 31173154. 

27 
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849877. 

28 
T. Tsujikawa, Global structure of the stationary solutions for the limiting system of a chemotaxisgrowth model, to appear in RIMS Kokyuroku, 2014. 

Go to top
