The fast signal diffusion limit in nonlinear chemotaxis systems

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March 2020, 25(3): 1109-1128. doi: 10.3934/dcdsb.2019211

The fast signal diffusion limit in nonlinear chemotaxis systems

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Received November 2018 Revised May 2019 Published March 2020 Early access September 2019

For

$ n\geq2 $

let

$ \mathit{\Omega }\subset {\mathbb{R}}^n $

be a bounded domain with smooth boundary as well as some nonnegative functions

$ 0\not \equiv u_0\in W^{1, \infty}(\mathit{\Omega }) $

and

$ v_0\in W^{1, \infty}(\mathit{\Omega }) $

. With

$ \varepsilon\in(0, 1) $

we want to know in which sense (if any!) solutions to the parabolic-parabolic system

$ \begin{equation*} \begin{cases} u_t = \nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u\nabla v) \;\;\; & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \varepsilon v_t = \mathit{\Delta } v -v+u & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 & \text{on} \ \partial\mathit{\Omega }\times\left(0, \infty \right), \\ u(\cdot, 0) = u_0, \ v(\cdot, 0) = v_0 & \text{in} \ \mathit{\Omega } \end{cases} \end{equation*} $

converge to those of the system where

$ \varepsilon = 0 $

and where the initial condition for

$ v $

has been removed. We will see in our theorem that indeed the solutions of these systems converge in a meaningful way if

$ m>1+\frac{n-2}{n} $

without the need for further conditions, e. g. on the size of

$ \left\|{{u_0}}\right\|_{L^p(\mathit{\Omega })} $

for some

$ p\in[1, \infty] $

.

Keywords: Chemotaxis, convergence, nonlinear parabolic equations, Keller-Segel system, global existence, boundedness.

Mathematics Subject Classification: 92C17, 35K55, 35B40.

Citation: Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211

References:

[1]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[2]	M. Freitag, Blow-up profiles and refined extensibility criteria in quasilinear Keller-Segel systems, Journal of Mathematical Analysis and Applications, 463 (2018), 964-988. doi: 10.1016/j.jmaa.2018.03.052.
[3]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[4]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 633–683.
[5]	W. Jäger and S. Luckhaus, On explosions to solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.
[6]	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.
[7]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI., 1968.
[8]	J. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2d Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189. doi: 10.1090/mcom/3250.
[9]	N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007.
[10]	N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic KellerSegel system, J. Math. Pures Appl. (9), 100 (2013), 748–767. doi: 10.1016/j.matpur.2013.01.020.
[11]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[12]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[13]	T. Nagai, T. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497. doi: 10.32917/hmj/1206124609.
[14]	Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34. doi: 10.4064/cm111-1-2.
[15]	M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Eq., 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.
[16]	T. Senba, Type ii blowup of solutions to a simplified Keller-Segel system in two dimensions, Nonlinear Anal., 66 (2007), 1817-1839. doi: 10.1016/j.na.2006.02.027.
[17]	Ph. Souplet and M. Winkler, Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3, M. Commun. Math. Phys., 367 (2019), 665-681. doi: 10.1007/s00220-018-3238-1.
[18]	T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.
[19]	T. Suzuki, Exclusion of boundary blowup for 2d chemotaxis system provided with Dirichlet boundary condition for the poisson part, J. Math. Pures Appl., 100 (2013), 347-367. doi: 10.1016/j.matpur.2013.01.004.
[20]	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.
[21]	Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263.
[22]	M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint.
[23]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[24]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system., Journal de Mathématiques Pures et Appliquees, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[25]	J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

show all references

References:

[1]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[2]	M. Freitag, Blow-up profiles and refined extensibility criteria in quasilinear Keller-Segel systems, Journal of Mathematical Analysis and Applications, 463 (2018), 964-988. doi: 10.1016/j.jmaa.2018.03.052.
[3]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[4]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 24 (1997), 633–683.
[5]	W. Jäger and S. Luckhaus, On explosions to solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.
[6]	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.
[7]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, Amer. math. Soc., Providence RI., 1968.
[8]	J. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2d Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189. doi: 10.1090/mcom/3250.
[9]	N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Annales de l'Institut Henri Poincaré, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007.
[10]	N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional parabolic KellerSegel system, J. Math. Pures Appl. (9), 100 (2013), 748–767. doi: 10.1016/j.matpur.2013.01.020.
[11]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[12]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
[13]	T. Nagai, T. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497. doi: 10.32917/hmj/1206124609.
[14]	Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34. doi: 10.4064/cm111-1-2.
[15]	M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Eq., 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.
[16]	T. Senba, Type ii blowup of solutions to a simplified Keller-Segel system in two dimensions, Nonlinear Anal., 66 (2007), 1817-1839. doi: 10.1016/j.na.2006.02.027.
[17]	Ph. Souplet and M. Winkler, Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3, M. Commun. Math. Phys., 367 (2019), 665-681. doi: 10.1007/s00220-018-3238-1.
[18]	T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.
[19]	T. Suzuki, Exclusion of boundary blowup for 2d chemotaxis system provided with Dirichlet boundary condition for the poisson part, J. Math. Pures Appl., 100 (2013), 347-367. doi: 10.1016/j.matpur.2013.01.004.
[20]	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.
[21]	Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263.
[22]	M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint.
[23]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[24]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system., Journal de Mathématiques Pures et Appliquees, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[25]	J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

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