• Previous Article
    Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal
  • DCDS-B Home
  • This Issue
  • Next Article
    A dimension splitting and characteristic projection method for three-dimensional incompressible flow
doi: 10.3934/dcdsb.2017199

Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

1. 

Dipartimento di Matematica e Informatica, Università di Cagliari, V. le Merello 92,09123. Cagliari, Italy

2. 

Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, United Kingdom

* Corresponding author: Giuseppe Viglialoro

Received  December 2016 Revised  April 2017 Published  July 2017

Fund Project: TEW would like to thank St John's College, Oxford and the Mathematical Biosciences Institute (MBI) at Ohio State University, for financially supporting this research through the National Science Foundation grant DMS 1440386 and BBSRC grant BKNXBKOO BK00.16

In this paper we study the chemotaxis-system
$\begin{equation*}\begin{cases}u_{t}=Δ u-χ \nabla · (u\nabla v)+g(u)&x∈ Ω, t>0, \\v_{t}=Δ v-v+u&x∈ Ω, t>0,\end{cases}\end{equation*}$
defined in a convex smooth and bounded domain
$Ω$
of
$\mathbb{R}^n$
,
$n≥q 1$
, with
$χ>0$
and endowed with homogeneous Neumann boundary conditions. The source
$g$
behaves similarly to the logistic function and satisfies
$g(s)≤q a -bs^α$
, for
$s≥q 0$
, with
$a≥q 0$
,
$b>0$
and
$α>1$
. Continuing the research initiated in [33], where for appropriate
$1 < p < α < 2$
and
$(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$
the global existence of very weak solutions
$(u,v)$
to the system (for any
$n≥q 1$
) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when
$n=3$
, we establish that
-for all
$τ>0$
an upper bound for
$\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$
can be prescribed in a such a way that
$(u,v)$
is bounded and Hölder continuous beyond
$τ$
;
-for all
$(u_0,v_0)$
, and sufficiently small ratio
$\frac{a}{b}$
, there exists a
$T>0$
such that
$(u,v)$
is bounded and Hölder continuous beyond
$T$
.
Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.
Citation: Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017199
References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015.

[2]

J. L. Aragón, R. A. Barrio, T. E. Woolley, R. E. Baker and P. K. Maini, Nonlinear effects on turing patterns: Time oscillations and chaos, Phys. Rev. E, 86 (2012), 026201.

[3]

N. BellomoA. BellouquidY. 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. doi: 10.1142/S021820251550044X.

[4]

J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theor. Biol. , 334 (2013), 1 – 12, URL http://www.sciencedirect.com/science/article/pii/S0022519313002646.

[5]

S. W. ChoS. KwakT. E. WoolleyM. J. LeeE. J. KimR. E. BakerH. J. KimJ. S. ShinC. TickleP. K. Maini and H. S. Jung, Interactions between shh, sostdc1 and wnt signaling and a new feedback loop for spatial patterning of the teeth, Development, 138 (2011), 1807-1816. doi: 10.1242/dev.056051.

[6]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis Nonlinearity, 21 (2008), 1057.

[7]

M. A. FarinaM. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Discret. Contin. Dyn. Syst. Suppl, (), 409-417. doi: 10.3934/proc.2015.0409.

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.

[9]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Eqns., 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, T. Am. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[11]

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.

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type in Translations of Mathematical Monographs, vol. 23, American Mathematical Society, 1988.

[13]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equations., 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016.

[14]

P. K. MainiT. E. WoolleyR. E. BakerE. A. Gaffney and S. S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496. doi: 10.1098/rsfs.2011.0113.

[15]

P. K. Maini, T. E. Woolley, E. A. Gaffney and R. E. Baker, The Once and Future Turing chapter 15: Biological pattern formation, Cambridge University Press, 2016.

[16]

M. MarrasS. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system, Discret. Contin. Dyn. Syst. Suppl, (), 809-916. doi: 10.3934/proc.2015.0809.

[17]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Method. Appl. Sci., 39 (2016), 2787-2798. doi: 10.1002/mma.3728.

[18]

M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulg. Sci., 69 (2016), 687-696.

[19]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications vol. 2, 3rd edition, Springer-Verlag, 2003.

[20]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.

[21]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.

[22]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvacioj., 44 (2001), 441-470.

[23]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676. doi: 10.1016/j.jmaa.2011.06.086.

[25]

L. -E. Persson and N. Samko, Inequalities and Convexity, in Operator Theory, Operator Algebras and Applications, Springer Basel, 2014,279–306.

[26]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[27]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. doi: 10.1016/j.jmaa.2011.02.041.

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Eqns., 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

[29]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Part. Diff. Eq., 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[30]

P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121.

[31]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232.

[32]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Int. Eqns., 29 (2016), 359-376.

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212. doi: 10.1016/j.jmaa.2016.02.069.

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535. doi: 10.1016/j.nonrwa.2016.10.001.

[35]

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.

[36]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[37]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part. Diff. Eq., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.

[38]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Method. Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.

[39]

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.

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[41]

T. E. Woolley, Spatiotemporal Behaviour of Stochastic and Continuum Models for Biological Signalling on Stationary and Growing Domains} PhD thesis, University of Oxford, 2011.

[42]

T. E. Woolley, 50 Visions of Mathematics chapter 48: Mighty Morphogenesis, Oxford Univ. Press, 2014.

[43]

T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini, Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation Phys. Rev. E, 84 (2011), 046216.

[44]

T. E. WoolleyR. E. BakerC. TickleP. K. Maini and M. Towers, Mathematical modelling of digit specification by a sonic hedgehog gradient, Dev. Dynam., 243 (2014), 290-298. doi: 10.1002/dvdy.24068.

show all references

References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London. Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015.

[2]

J. L. Aragón, R. A. Barrio, T. E. Woolley, R. E. Baker and P. K. Maini, Nonlinear effects on turing patterns: Time oscillations and chaos, Phys. Rev. E, 86 (2012), 026201.

[3]

N. BellomoA. BellouquidY. 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. doi: 10.1142/S021820251550044X.

[4]

J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theor. Biol. , 334 (2013), 1 – 12, URL http://www.sciencedirect.com/science/article/pii/S0022519313002646.

[5]

S. W. ChoS. KwakT. E. WoolleyM. J. LeeE. J. KimR. E. BakerH. J. KimJ. S. ShinC. TickleP. K. Maini and H. S. Jung, Interactions between shh, sostdc1 and wnt signaling and a new feedback loop for spatial patterning of the teeth, Development, 138 (2011), 1807-1816. doi: 10.1242/dev.056051.

[6]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis Nonlinearity, 21 (2008), 1057.

[7]

M. A. FarinaM. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Discret. Contin. Dyn. Syst. Suppl, (), 409-417. doi: 10.3934/proc.2015.0409.

[8]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.

[9]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Eqns., 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, T. Am. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[11]

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.

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type in Translations of Mathematical Monographs, vol. 23, American Mathematical Society, 1988.

[13]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equations., 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016.

[14]

P. K. MainiT. E. WoolleyR. E. BakerE. A. Gaffney and S. S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496. doi: 10.1098/rsfs.2011.0113.

[15]

P. K. Maini, T. E. Woolley, E. A. Gaffney and R. E. Baker, The Once and Future Turing chapter 15: Biological pattern formation, Cambridge University Press, 2016.

[16]

M. MarrasS. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system, Discret. Contin. Dyn. Syst. Suppl, (), 809-916. doi: 10.3934/proc.2015.0809.

[17]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Math. Method. Appl. Sci., 39 (2016), 2787-2798. doi: 10.1002/mma.3728.

[18]

M. Marras and G. Viglialoro, Blow-up time of a general Keller-Segel system with source and damping terms, C. R. Acad. Bulg. Sci., 69 (2016), 687-696.

[19]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications vol. 2, 3rd edition, Springer-Verlag, 2003.

[20]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.

[21]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Theory Methods Appl., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.

[22]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvacioj., 44 (2001), 441-470.

[23]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.

[24]

L. E. Payne and J. C. Song, Lower bounds for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676. doi: 10.1016/j.jmaa.2011.06.086.

[25]

L. -E. Persson and N. Samko, Inequalities and Convexity, in Operator Theory, Operator Algebras and Applications, Springer Basel, 2014,279–306.

[26]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[27]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529. doi: 10.1016/j.jmaa.2011.02.041.

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Eqns., 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

[29]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Part. Diff. Eq., 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[30]

P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121.

[31]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232.

[32]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Diff. Int. Eqns., 29 (2016), 359-376.

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212. doi: 10.1016/j.jmaa.2016.02.069.

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535. doi: 10.1016/j.nonrwa.2016.10.001.

[35]

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.

[36]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[37]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part. Diff. Eq., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.

[38]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Method. Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.

[39]

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.

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[41]

T. E. Woolley, Spatiotemporal Behaviour of Stochastic and Continuum Models for Biological Signalling on Stationary and Growing Domains} PhD thesis, University of Oxford, 2011.

[42]

T. E. Woolley, 50 Visions of Mathematics chapter 48: Mighty Morphogenesis, Oxford Univ. Press, 2014.

[43]

T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini, Stochastic reaction and diffusion on growing domains: Understanding the breakdown of robust pattern formation Phys. Rev. E, 84 (2011), 046216.

[44]

T. E. WoolleyR. E. BakerC. TickleP. K. Maini and M. Towers, Mathematical modelling of digit specification by a sonic hedgehog gradient, Dev. Dynam., 243 (2014), 290-298. doi: 10.1002/dvdy.24068.

Figure 1.  Simulations of system (45) in one dimension with varying value of $\alpha$, given beneath each subfigure. Each subfigure contains the system evaluated at the time points $t=1$, 10, 50 and 100. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1000 equally spaced points
Figure 2.  Simulations of system (45) in one dimension. The simulations are nearly identical to those seen in Figure 1(a). However, each simulation involves a single parameter change. Specifically, in (a) a larger initial condition for $u$ was used (100 was added to the mean); in (b) the parameter $b$ was reduced to 0.2; Finally, in (c) the spatial solution domain has been reduced from 10 to 1
Figure 3.  Simulations of system (45) in two dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was triangulated into 24, 968 finite elements. The figure inset of (b) shows the full extent of the peak, which is growing without bound
Figure 4.  Simulations of system (45) illustrating the density of $u$ in three dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1, 139, 254 voxel elements. Apart from the light grey ball illustrating the boundary of the solution domain the images illustrate isosurfaces of the solution (i.e. surface that represent points of a constant value, thus, they are the three-dimensional analogue of contours). In Figure (a) there are five isosurfaces of value 1, 1.25, 1.5 1.75 and 2, coloured, yellow, green, blue, red and black, respectively. In Figure (b) there are three isosurfaces of value 1, 10, and $10^6$, coloured, yellow, blue and black, respectively
[1]

Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789

[2]

Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503

[3]

Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018

[4]

Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299

[5]

Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034

[6]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268

[7]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125

[8]

Michael Winkler. Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2777-2793. doi: 10.3934/dcdsb.2017135

[9]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-25. doi: 10.3934/dcdsb.2018180

[10]

Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132

[11]

Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147

[12]

Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216

[13]

Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737

[14]

Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094

[15]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150

[16]

Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031

[17]

Giuseppina Autuori, Patrizia Pucci. Kirchhoff systems with nonlinear source and boundary damping terms. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1161-1188. doi: 10.3934/cpaa.2010.9.1161

[18]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078

[19]

Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure & Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587

[20]

Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035

2017 Impact Factor: 0.972

Article outline

Figures and Tables

[Back to Top]