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

Giuseppe Viglialoro; Thomas E. Woolley

doi:10.3934/dcdsb.2017199

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October 2018, 23(8): 3023-3045. doi: 10.3934/dcdsb.2017199

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

Giuseppe Viglialoro ^1,, and Thomas E. Woolley ^2,

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 October 2018 Early access 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

$Ω$

$\mathbb{R}^n$

$n≥ 1$

, with

$χ>0$

and endowed with homogeneous Neumann boundary conditions. The source

$g$

behaves similarly to the logistic function and satisfies

$g(s)≤ a -bs^α$

, for

$s≥ 0$

, with

$a≥ 0$

$b>0$

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

$n≥ 1$

) 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.

Keywords: Nonlinear parabolic systems, chemotaxis, logistic source, boundedness, eventual smoothness.

Mathematics Subject Classification: Primary: 35B40, 35K55, 35B65, 92C17; Secondary: 35Q92.

Citation: Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199

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.

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

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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

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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

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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

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2021 Impact Factor: 1.497

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2021 Impact Factor: 1.497

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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