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doi: 10.3934/dcdss.2022045
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Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion

Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educación a Distancia, 28040 Madrid, Spain

Dedicated to Professor Georg Hetzer on the Occasion of his 75th Birthday

Received  August 2021 Revised  December 2021 Early access March 2022

Fund Project: The author is supported by Ministerio de Ciencia e Innovación, Spain, under grant number MTM2017-83391-P

We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a
$ N $
-dimensional unit ball "
$ B $
" describing the behavior of a biological species "
$ u $
" and a chemical stimuli "
$ v $
". The system presents a sub-linear dependence of "
$ \nabla v $
" in the chemotactic coefficient and a nonlinear diffusive term. The evolution of
$ u $
is described by the equation
$ u_t - \Delta u^m = - div (\chi u |\nabla v|^{p-2} \nabla v), \quad \mbox{ for } \ m >2, \quad p \in ( 1,2), \quad N \geq 1 $
for a positive constant
$ \chi $
. The concentration of the chemical substance
$ v $
satisfies the linear elliptic equation
$ - \Delta v = u - \frac{1}{|B|} \int_{B} u_0dx. $
We consider the radially symmetric case and we prove the local existence of weak solutions for the mass accumulation function under assumption
$ - \frac{1}{m}+ \frac{1}{N} + 1-\frac{pm}{4(m-1)} \geq 0, $
for radial and regular initial data. Additionally, if the constrain
$ \frac{m }{m- 2} \left[ \frac{pm}{2(m-1)}-1\right] \leq 1 $
is satisfied, the solution globally exists in time.
Citation: J. Ignacio Tello. Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022045
References:
[1]

W. Alt, Orientation of cells migrating in a chemotactic gradient, in Biological Growth and Spread, Lecture Notes in Biomath 38, Springer-Verlag, New York, (1980), 353–366.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands 1976.

[3]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci., 20 (2010), 1179-1207.  doi: 10.1142/S0218202510004568.

[4]

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.

[5]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations, 42 (2017), 436-473.  doi: 10.1080/03605302.2016.1277237.

[6]

N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.  doi: 10.1090/btran/17.

[7]

A. BianchiK. J. Painter and J. A. Sherratt, A mathematical model for lymphangiogenesis in normal and diabetic wounds, J. Theoret. Biol., 383 (2015), 61-86.  doi: 10.1016/j.jtbi.2015.07.023.

[8]

A. BianchiK. J. Painter and J. A. Sherratt, Spatio-temporal models of lymphangiogenesisin wound healing, Bull. Math. Biol., 78 (2016), 1904-1941.  doi: 10.1007/s11538-016-0205-x.

[9]

P. BilerE. E. Espejo and I. Guerra, Blow-up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[10]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift–diffusion systems, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), 5186-5193.  doi: 10.1016/j.na.2012.04.035.

[11]

L. Boccardo and L. Orsina, Sublinear elliptic systems with a convection term, Comm. Partial Differential Equations, 45 (2020), 690-713.  doi: 10.1080/03605302.2020.1712417.

[12]

Y. ChiyodaM. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta Appl. Math., 167 (2020), 231-259.  doi: 10.1007/s10440-019-00275-z.

[13]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[14]

W. CintraC. Morales-Rodrigo and A. Suárez, Coexistence states in a cross-diffusion system of a predator-prey model with predator satiation term, Math. Models Methods Appl. Sci., 28 (2018), 2131-2159.  doi: 10.1142/S0218202518400109.

[15]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356.  doi: 10.1016/j.aml.2011.09.013.

[16]

C. ConcaE. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $R^2$, European J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258.

[17]

E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[18]

E. E. EspejoA. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288. 

[19]

E. E. EspejoA. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462. 

[20]

E. E. Espejo ArenasA. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029.

[21]

E. GalakhovO. Salieva and J. I. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[22]

G. HetzerT. Nguyen and W. Shen, A-stability of global attractors of competition diffusion systems, J. Dyn. Diff. Equat., 22 (2010), 533-561.  doi: 10.1007/s10884-010-9187-9.

[23]

G. Hetzer and W. Shen, Two species competition with an inhibitor involved, Discrete Contin. Dyn. Syst., 12 (2005), 39-57.  doi: 10.3934/dcds.2005.12.39.

[24]

G. Hetzer and W. Shen, Convergence in almost periodic competition diffusion systems, J. Math. Anal. Appl., 262 (2001), 307-338.  doi: 10.1006/jmaa.2001.7582.

[25]

G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic / nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.

[26]

G. Hetzer and L. Tello, A convergence theorem for a two-species competition system with slow diffusion, Electron. J. Differ. Equ. Conf., 22 (2015), 47-51. 

[27]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[28]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[29]

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.  doi: 10.1007/s00332-010-9082-x.

[30]

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.

[31]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[32]

A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, Singapore, World-Scientific, 2003. doi: 10.1142/5129.

[33]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Paris, Dunod, 1969.

[34]

M. Negreanu, Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3335-3356.  doi: 10.3934/dcdsb.2020064.

[35]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[36]

M. Negreanu and J. I. Tello, On a parabolic-elliptic system with gradient dependent chemotactic coefficient, J. Differential Equations, 265 (2018), 733-751.  doi: 10.1016/j.jde.2018.01.040.

[37]

J. F. PadialP. Takáč and L. Tello, An antimaximum principle for a degenerate parabolic problem, Adv. Differential Equations, 15 (2010), 601-648. 

[38]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on ${{\mathbb{R}}}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.

[39]

R. B. Salako and W. Shen, Existence of traveling wave solutions of parabolic-parabolic chemotaxis systems, Nonlinear Anal. Real World Appl., 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[40]

J. I. Tello, Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficient, Comm. Partial Differential Equations, 47 (2022), 307-345.  doi: 10.1080/03605302.2021.1975132.

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[42]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.

[43]

G. Viglialoro, Global in time and bounded solutions to a parabolic-elliptic chemotaxis system with nonlinear diffusion and signal-dependent sensitivity, Appl. Math. Optim., 83 (2021), 979-1004.  doi: 10.1007/s00245-019-09575-0.

[44]

M. Winkler, A critical blow-up exponent for flux limitation in a Keller-Segel system, Indiana Univ. Math. J., To appear.

[45]

G. Wolansky, Multi-components chemotactic system in absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.

show all references

References:
[1]

W. Alt, Orientation of cells migrating in a chemotactic gradient, in Biological Growth and Spread, Lecture Notes in Biomath 38, Springer-Verlag, New York, (1980), 353–366.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands 1976.

[3]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci., 20 (2010), 1179-1207.  doi: 10.1142/S0218202510004568.

[4]

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.

[5]

N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations, 42 (2017), 436-473.  doi: 10.1080/03605302.2016.1277237.

[6]

N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.  doi: 10.1090/btran/17.

[7]

A. BianchiK. J. Painter and J. A. Sherratt, A mathematical model for lymphangiogenesis in normal and diabetic wounds, J. Theoret. Biol., 383 (2015), 61-86.  doi: 10.1016/j.jtbi.2015.07.023.

[8]

A. BianchiK. J. Painter and J. A. Sherratt, Spatio-temporal models of lymphangiogenesisin wound healing, Bull. Math. Biol., 78 (2016), 1904-1941.  doi: 10.1007/s11538-016-0205-x.

[9]

P. BilerE. E. Espejo and I. Guerra, Blow-up in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.  doi: 10.3934/cpaa.2013.12.89.

[10]

P. Biler and I. Guerra, Blowup and self-similar solutions for two-component drift–diffusion systems, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), 5186-5193.  doi: 10.1016/j.na.2012.04.035.

[11]

L. Boccardo and L. Orsina, Sublinear elliptic systems with a convection term, Comm. Partial Differential Equations, 45 (2020), 690-713.  doi: 10.1080/03605302.2020.1712417.

[12]

Y. ChiyodaM. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta Appl. Math., 167 (2020), 231-259.  doi: 10.1007/s10440-019-00275-z.

[13]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[14]

W. CintraC. Morales-Rodrigo and A. Suárez, Coexistence states in a cross-diffusion system of a predator-prey model with predator satiation term, Math. Models Methods Appl. Sci., 28 (2018), 2131-2159.  doi: 10.1142/S0218202518400109.

[15]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356.  doi: 10.1016/j.aml.2011.09.013.

[16]

C. ConcaE. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $R^2$, European J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258.

[17]

E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[18]

E. E. EspejoA. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288. 

[19]

E. E. EspejoA. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462. 

[20]

E. E. Espejo ArenasA. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029.

[21]

E. GalakhovO. Salieva and J. I. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[22]

G. HetzerT. Nguyen and W. Shen, A-stability of global attractors of competition diffusion systems, J. Dyn. Diff. Equat., 22 (2010), 533-561.  doi: 10.1007/s10884-010-9187-9.

[23]

G. Hetzer and W. Shen, Two species competition with an inhibitor involved, Discrete Contin. Dyn. Syst., 12 (2005), 39-57.  doi: 10.3934/dcds.2005.12.39.

[24]

G. Hetzer and W. Shen, Convergence in almost periodic competition diffusion systems, J. Math. Anal. Appl., 262 (2001), 307-338.  doi: 10.1006/jmaa.2001.7582.

[25]

G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic / nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.

[26]

G. Hetzer and L. Tello, A convergence theorem for a two-species competition system with slow diffusion, Electron. J. Differ. Equ. Conf., 22 (2015), 47-51. 

[27]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[28]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[29]

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.  doi: 10.1007/s00332-010-9082-x.

[30]

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.

[31]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[32]

A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, Singapore, World-Scientific, 2003. doi: 10.1142/5129.

[33]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Paris, Dunod, 1969.

[34]

M. Negreanu, Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3335-3356.  doi: 10.3934/dcdsb.2020064.

[35]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009.

[36]

M. Negreanu and J. I. Tello, On a parabolic-elliptic system with gradient dependent chemotactic coefficient, J. Differential Equations, 265 (2018), 733-751.  doi: 10.1016/j.jde.2018.01.040.

[37]

J. F. PadialP. Takáč and L. Tello, An antimaximum principle for a degenerate parabolic problem, Adv. Differential Equations, 15 (2010), 601-648. 

[38]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on ${{\mathbb{R}}}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.

[39]

R. B. Salako and W. Shen, Existence of traveling wave solutions of parabolic-parabolic chemotaxis systems, Nonlinear Anal. Real World Appl., 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[40]

J. I. Tello, Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficient, Comm. Partial Differential Equations, 47 (2022), 307-345.  doi: 10.1080/03605302.2021.1975132.

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[42]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.

[43]

G. Viglialoro, Global in time and bounded solutions to a parabolic-elliptic chemotaxis system with nonlinear diffusion and signal-dependent sensitivity, Appl. Math. Optim., 83 (2021), 979-1004.  doi: 10.1007/s00245-019-09575-0.

[44]

M. Winkler, A critical blow-up exponent for flux limitation in a Keller-Segel system, Indiana Univ. Math. J., To appear.

[45]

G. Wolansky, Multi-components chemotactic system in absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.

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Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure and Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719

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