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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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