# American Institute of Mathematical Sciences

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

## The fast signal diffusion limit in nonlinear chemotaxis systems

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
 $p\in[1, \infty]$
.
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.

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##### 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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