×

zbMATH — the first resource for mathematics

Asymptotic profile of solutions to a certain chemotaxis system. (English) Zbl 1435.92010
The authors study the chemotaxis system with consumption \[ \begin{cases} \partial_t n = \Delta n - \nabla \cdot (n \nabla c), & \quad (x,t) \in \mathbb{R}^2 \times (0,\infty), \\ \partial_t c = \Delta c - nc, & \quad (x,t) \in \mathbb{R}^2 \times (0,\infty), \end{cases} \] endowed with positive initial data \((n_0,c_0) \in (L^1(\mathbb{R}^2) \cap L^2(\mathbb{R}^2)) \times (L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2))\) such that \(\nabla \sqrt{c_0} \in L^2(\mathbb{R}^2)\). Here \(n\) is the cell density and \(c\) the concentration of a chemical, e.g. oxygen, and it is assumed that the cells are attracted by the chemical and consume the latter.
The existence of a unique global solution has been shown in [Q. Zhang and X. Zheng, SIAM J. Math. Anal. 46, No. 4, 3078–3105 (2014; Zbl 1444.35011)]. The authors now prove the following large time behavior:
Denoting by \(G(x,t) := (4\pi t)^{-1} \exp(\frac{|x|^2}{4t})\) the heat kernel and by \(e^{t\Delta}\) the heat semigroup on \(\mathbb{R}^2\), the authors show that \[ \| n(t) - e^{t\Delta} n_0 \|_{L^p(\mathbb{R}^2)} \le C_p t^{-\frac{3}{2} + \frac{1}{p}} \quad \forall t>0, \] \[ \| c(t) - M_c G(t) \|_{L^q(\mathbb{R}^2)} = o \left(t^{-1 + \frac{1}{q}} \right) \quad \mbox{as } t \to \infty \] hold for any \(p \in [1,\infty]\) and \(q \in [1,\infty)\) and that, if in addition \(n_0 \in L^1(\mathbb{R}^2; (1+|x|^2)dx)\), then also \[\| n(t) - M_n G(t) \|_{L^p(\mathbb{R}^2)} \le C_p t^{-\frac{3}{2} + \frac{1}{p}} \quad \forall t>0\] is satisfied, where \(M_n\) is the initial cell mass.
The proofs in particular rely on appropriate energy estimates and properties of the heat semigroup.
MSC:
92C17 Cell movement (chemotaxis, etc.)
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. Biler; M. Guedda; G. Karch, Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation, J. Evol. Equ., 4, 75-97 (2004) · Zbl 1056.35024
[2] M. Chae; K. Kang; J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39, 1205-1235 (2014) · Zbl 1304.35481
[3] T. Hillen; K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1161.92003
[4] E. F. Keller; L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26, 399-415 (1970) · Zbl 1170.92306
[5] H. Kozono; M. Miura; Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270, 1663-1683 (2016) · Zbl 1343.35069
[6] Y. Li; Y. Li, Global boundedness of solutions for the chemotaxis-Navier-Stokes system in \(\mathbb{R}^2\), J. Differential Equations, 261, 6570-6613 (2016) · Zbl 1354.35071
[7] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381, 521-529 (2011) · Zbl 1225.35118
[8] Y. Tao; M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252, 2520-2543 (2012) · Zbl 1268.35016
[9] I. Tuval; L. Cisneros; C. Dombrowski; C. W. Wolgemuth; J. O. Kessler; R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA, 102, 2277-2282 (2005) · Zbl 1277.35332
[10] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37, 319-351 (2012) · Zbl 1236.35192
[11] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211, 455-487 (2014) · Zbl 1293.35220
[12] Q. Zhang; X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46, 3078-3105 (2014) · Zbl 1444.35011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.