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A new criterion to a two-chemical substances chemotaxis system with critical dimension

The first author is supported by Alexander von Humboldt Foundation, NSF (No.11501207), Postdoctoral Science Foundation of China (No. 2016M600812) and Special financial aid to postdoctor research fellow (No.2017T100768). The second author is supported by NSF (No.11701453) and Postdoctoral Science Foundation of China (No. 2016M600811).
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  • We mainly investigate the global boundedness of the solution to the following system,

    $\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$

    under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain $Ω\subset \mathbb{R}^n$ with critical space dimension $n = 4$. This problem has been considered by K. Fujie and T. Senba in [5]. They proved that for the symmetric case the condition $\int_\Omega {u_0 < \frac{(8π)^2}{χ}} $ yields global boundedness, where $u_0$ is the instal data for $u$. In this paper, inspired by some new techniques established in [3], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [5].

    Mathematics Subject Classification: Primary: 35B45, 35K45; Secondary: 35Q92, 92C17.


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    [2] N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [3] X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv:1707.09235.
    [4] X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model Zeitschrift Für Angewandte Mathematik Und Physik, 67 (2016). doi: 10.1007/s00033-015-0601-3.
    [5] K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differetial Equations, 37 (2017), 61-83.  doi: 10.1016/j.jde.2017.02.031.
    [6] M. Hieber and J. Prüss, Heat kernels and maximal lp-lq estimate for parabolic evolution equations, Comm. Partial Differential Equaitons, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.
    [7] T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.
    [8] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jber. DMV, 105 (2003), 103-165. 
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