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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 |
$\begin{equation*}\begin{cases}u_{t}=Δ u-χ \nabla · (u\nabla v)+g(u)&x∈ Ω, t>0, \\v_{t}=Δ v-v+u&x∈ Ω, t>0,\end{cases}\end{equation*}$ |
$Ω$ |
$\mathbb{R}^n$ |
$n≥ 1$ |
$χ>0$ |
$g$ |
$g(s)≤ a -bs^α$ |
$s≥ 0$ |
$a≥ 0$ |
$b>0$ |
$α>1$ |
$1 < p < α < 2$ |
$(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$ |
$(u,v)$ |
$n≥ 1$ |
$n=3$ |
$τ>0$ |
$\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$ |
$(u,v)$ |
$τ$ |
$(u_0,v_0)$ |
$\frac{a}{b}$ |
$T>0$ |
$(u,v)$ |
$T$ |
References:
[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), 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. 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.
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.
doi: 10.1016/j.jtbi.2013.05.033. |
[5] |
S. W. Cho, S. Kwak, T. E. Woolley, M. J. Lee, E. J. Kim, R. E. Baker, H. J. Kim, J. S. Shin, C. Tickle, P. 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.
doi: 10.1088/0951-7715/21/5/009. |
[7] |
M. A. Farina, M. 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. Maini, T. E. Woolley, R. E. Baker, E. 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. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolicparabolic Keller–Segel system, Discret. Contin. Dyn. Syst. Suppl, 809–916.
doi: 10.3934/proc.2015.0809. |
[17] |
M. Marras, S. 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. Osaki, T. Tsujikawa, A. 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.
doi: 10.1007/978-3-0348-0816-3_17. |
[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.
doi: 10.1103/PhysRevE.84.046216. |
[44] |
T. E. Woolley, R. E. Baker, C. Tickle, P. 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. 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), 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. 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.
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.
doi: 10.1016/j.jtbi.2013.05.033. |
[5] |
S. W. Cho, S. Kwak, T. E. Woolley, M. J. Lee, E. J. Kim, R. E. Baker, H. J. Kim, J. S. Shin, C. Tickle, P. 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.
doi: 10.1088/0951-7715/21/5/009. |
[7] |
M. A. Farina, M. 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. Maini, T. E. Woolley, R. E. Baker, E. 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. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolicparabolic Keller–Segel system, Discret. Contin. Dyn. Syst. Suppl, 809–916.
doi: 10.3934/proc.2015.0809. |
[17] |
M. Marras, S. 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. Osaki, T. Tsujikawa, A. 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.
doi: 10.1007/978-3-0348-0816-3_17. |
[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.
doi: 10.1103/PhysRevE.84.046216. |
[44] |
T. E. Woolley, R. E. Baker, C. Tickle, P. 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. |




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