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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.
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
Full Text: DOI
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