American Institute of Mathematical Sciences

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November  2018, 38(11): 5943-5961. doi: 10.3934/dcds.2018258

Global existence and boundedness in a chemorepulsion system with superlinear diffusion

Marcel Freitag ,

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

* Corresponding author: Marcel Freitag

Received  May 2018 Revised  June 2018 Published  August 2018

Fund Project: The author is supported by the Deutsche Forschungsgemeinschaft

In a bounded domain
$\Omega\subset\mathbb{R}^n$
, where
$n\ge 3$
, we consider the quasilinear parabolic-parabolic Keller-Segel system
$\begin{equation*}\begin{cases}u_t = \nabla\cdot({D(u)\nabla u+u\nabla v}) \;\;\; &\text{in}\ \Omega\times(0,\infty)\\v_t = \Delta v-v+u &\text{in}\ \Omega\times(0,\infty)\end{cases}\end{equation*}$
with homogeneous Neumann boundary conditions. We will find that the condition
$D(u)\geq Cu^{m-1}$
 suffices to prove the uniqueness and global existence of solutions along with their boundedness if
$D(0)>0$
 and
$m>1+\frac{(n-2)(n-1)}{n^2}$
 which is a very different result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diffusion (
$D(0) = 0$
) and for the same parameters, locally bounded global weak solutions will be established.
Keywords: Classical solutions, blow-up, nonlinear parabolic equations, critical exponents, cell movement (chemotaxis, etc.).
Mathematics Subject Classification: 35A09, 35B44, 35K35, 35B33, 92C17.
Citation: Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258
References:
[1]

N. Bellomo, A. Belloquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

[2]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378. Google Scholar

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discr. Cont. Dyn. Syst. A, 35 (2005), 1891-1904. doi: 10.3934/dcds.2015.35.1891. Google Scholar

[4]

T. Cieślak, Quasilinear nonuniformly parabolic system modeling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080. Google Scholar

[5]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel systems in higher dimensions, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jde.2012.01.045. Google Scholar

[6]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differ. Eq., 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. Google Scholar

[7]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117. doi: 10.4064/bc81-0-7. Google Scholar

[8]

X., Fu, L. H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial systems with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.Google Scholar

[9]

K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151. Google Scholar

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. Google Scholar

[11]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683. Google Scholar

[12]

T. Hillen and K. J. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

D. Horstmann and M. Winkler, Boundedness vs! blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2015), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[16]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[17]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar

[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, 1968. Google Scholar

[19]

J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, Journal of Differential Equations, 262 (2017), 4052-4084. doi: 10.1016/j.jde.2016.12.007. Google Scholar

[20]

H. A. Levine and B. D. Sleeman, Partial differential equations of chemotaxis and angiogenesis. Applied mathematical analysis in the last century, Math. Meth. Appl. Sci., 24 (2001), 405-426. doi: 10.1002/mma.212. Google Scholar

[21]

J. F Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion, Physica A, 392 (2013), 5644-5662. doi: 10.1016/j.physa.2013.07.022. Google Scholar

[22]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré C, Analyse non liné aire, 31 (2013), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[23]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[24]

T. Nagai, T. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar

[25]

L. Nirenberg, An extended interpolation inequality, Annali della Scuola Normale Superiore di Pisa, Classe die Scienze 3e sé rie, 20 (1966), 733-737. Google Scholar

[26]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2011), 441-469. Google Scholar

[27]

T. Senba and T. Suzuki, A quasi-linear system of chemotaxis, Abstr. Appl. Anal., 2006 (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061. Google Scholar

[28]

J. Simon, Compact sets in the space $L^p\left(0, T;B \right)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[29]

C. Stinner, C. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Mod. Meth. Appl. Sci., 26 (2016), 2163-2201. doi: 10.1142/S021820251640011X. Google Scholar

[30]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[31]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705. Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. Google Scholar

[33]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wohlgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Accad. Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. Google Scholar

[34]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar

[35]

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

[36]

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

[37]

D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Mod. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106. Google Scholar

show all references

References:
[1]

N. Bellomo, A. Belloquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

[2]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378. Google Scholar

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discr. Cont. Dyn. Syst. A, 35 (2005), 1891-1904. doi: 10.3934/dcds.2015.35.1891. Google Scholar

[4]

T. Cieślak, Quasilinear nonuniformly parabolic system modeling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jmaa.2006.03.080. Google Scholar

[5]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel systems in higher dimensions, J. Math. Anal. Appl., 326 (2007), 1410-1426. doi: 10.1016/j.jde.2012.01.045. Google Scholar

[6]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differ. Eq., 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. Google Scholar

[7]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117. doi: 10.4064/bc81-0-7. Google Scholar

[8]

X., Fu, L. H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial systems with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.Google Scholar

[9]

K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151. Google Scholar

[10]

H. Gajewski and K. Zacharias, Global behaviour of a reaction diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. Google Scholar

[11]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683. Google Scholar

[12]

T. Hillen and K. J. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

D. Horstmann and M. Winkler, Boundedness vs! blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2015), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[16]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[17]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar

[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, 1968. Google Scholar

[19]

J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, Journal of Differential Equations, 262 (2017), 4052-4084. doi: 10.1016/j.jde.2016.12.007. Google Scholar

[20]

H. A. Levine and B. D. Sleeman, Partial differential equations of chemotaxis and angiogenesis. Applied mathematical analysis in the last century, Math. Meth. Appl. Sci., 24 (2001), 405-426. doi: 10.1002/mma.212. Google Scholar

[21]

J. F Leyva, C. Málaga and R. G. Plaza, The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion, Physica A, 392 (2013), 5644-5662. doi: 10.1016/j.physa.2013.07.022. Google Scholar

[22]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré C, Analyse non liné aire, 31 (2013), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar

[23]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[24]

T. Nagai, T. Senba and K. Yoshida, Applications of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar

[25]

L. Nirenberg, An extended interpolation inequality, Annali della Scuola Normale Superiore di Pisa, Classe die Scienze 3e sé rie, 20 (1966), 733-737. Google Scholar

[26]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2011), 441-469. Google Scholar

[27]

T. Senba and T. Suzuki, A quasi-linear system of chemotaxis, Abstr. Appl. Anal., 2006 (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061. Google Scholar

[28]

J. Simon, Compact sets in the space $L^p\left(0, T;B \right)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[29]

C. Stinner, C. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Mod. Meth. Appl. Sci., 26 (2016), 2163-2201. doi: 10.1142/S021820251640011X. Google Scholar

[30]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X. Google Scholar

[31]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705. Google Scholar

[32]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. Google Scholar

[33]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wohlgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Accad. Sci. USA, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102. Google Scholar

[34]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar

[35]

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

[36]

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

[37]

D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Mod. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106. Google Scholar

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