Finite-time blow-up in a quasilinear chemotaxis system with an external signal consumption
Keywords
Finite-time blow-up, chemotaxis, external signal consumptionAbstract
This paper deals with a quasilinear chemotaxis system with an external signal consumption \begin{equation*}%\label{1a} \begin{cases} u_t=\nabla\cdot(\varphi(u)\nabla u)-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v+u-g(x), &(x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a ball $\Omega\subset \mathbb{R}^{n}$, where $\varphi(u)$ is a nonlinear diffusion function and $g(x)$ is an external signal consumption. Under suitable assumptions on the functions $\varphi$ and $g$, it is proved that there exists initial data such that the solution of the above system blows up in finite time.References
N.D. Alikakos, bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827–868.
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), 1663–1763.
T. Black, Boundedness in a Keller–Segel system with external signal production, J. Math. Anal. Appl. 446 (2017), 436–455.
T. Black, Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant, Nonlinearity 29 (2016), 1865–1886.
J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller–Segel system, Nonlinear Anal. 75 (2012), 5215–5228.
X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Math. Meth. Appl. Sci. 37 (2014), 2326–2330.
T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl. 326 (2007), 1410–1426.
T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations252 (2012), 5832–5851.
T. Cieślak 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.
T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008), 1057–1076.
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
T. Hillen and K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217.
D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller–Segel model, J. Math. Biol. 44 (2002), 463–478.
D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci. 21 (2011), 231–270.
D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I, Jahresber. Dtsch. Math. -Ver. 105 (2003), 103–165.
D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences II, Jahresber. Dtsch. Math.-Ver. 106 (2004), 51–69.
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math. 12 (2001), 159–177.
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005), 52–107.
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824.
E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399–415.
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.
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 20 (1966), 733–737.
K.J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002), 501–543.
Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci. 1 (2013), 1–36.
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Differential Equations 252 (2012), 692–715.
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.
J.I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations 32 (2007), no. 6, 849–877.
J.I. Tello and M. Winkler, Reduction of critical mass in a chemotaxis system by external application of chemoattractant, Ann. Sci. Norm. Super. Pisa Cl. Sci. 12 (2013), 833–862.
W. Wagner, et al., Hematopoietic progenitor cells and cellular microenvironment: behavioral and molecular changes upon interaction, Stem Cells 23 (2015), 1180–1191.
L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations 256 (2014), 1847–1872.
Z.A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom. 5 (2010), 173–190.
Z.A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos 17 (2007), 037108.
Z.A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity 24 (2011), 3279–3297.
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.
M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl. 384 (2011), 261–272.
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl. 100 (2013), 748–767.
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations 248 (2010), 2889–2905.
M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations 35 (2010), 1516–1537.
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr. 283 (2010), 1664–1673.
M. Winkler, Does a ‘volume-filling effect’ always prevent chemotactic collapse?, Math. Methods Appl. Sci. 33 (2010), 12–24.
M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl. 348 (2008), 708–729.
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.
D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal. 73 (2010), 338–349.
P. Zheng and C. Mu, Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source, Math. Nachr. 288 (2015), 710–720.
P. Zheng, C. Mu and X. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst. 35 (2015), 2299–2323.
P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl. 424 (2015), 509–522.
P. Zheng, C. Mu, L. Wang and L. Li, Boundedness and asymptotic behavior in a fully parabolic chemotaxis-growth system with signal-dependent sensitivity, J. Evol. Equ. 17 (2017), 909–929.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0