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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
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##### References:
  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  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  T. Hillen; K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1161.92003  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  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  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  Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381, 521-529 (2011) · Zbl 1225.35118  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  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  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  M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211, 455-487 (2014) · Zbl 1293.35220  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
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