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] $
.
Citation: Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. LiuL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[12]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[13]

T. NagaiT. 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.  Google Scholar

[14]

Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.  doi: 10.4064/cm111-1-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263. Google Scholar

[22]

M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. LiuL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[12]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.   Google Scholar

[13]

T. NagaiT. 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.  Google Scholar

[14]

Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.  doi: 10.4064/cm111-1-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Preprint, 2018, arXiv: 1805.05263. Google Scholar

[22]

M. Winkler, Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
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