November  2018, 23(9): 3717-3721. doi: 10.3934/dcdsb.2018074

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

Department of Applied Mathematics, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, Shaanxi, China

* Corresponding author: mybxl110@163.com

Received  September 2017 Revised  November 2017 Published  January 2018

Fund Project: 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)

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].
Citation: Xueli Bai, Suying Liu. A new criterion to a two-chemical substances chemotaxis system with critical dimension. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3717-3721. doi: 10.3934/dcdsb.2018074
References:
[1]

N. D. Alikakos, Lp-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar

[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. Google Scholar

[3]

X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv:1707.09235.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jber. DMV, 105 (2003), 103-165. Google Scholar

show all references

References:
[1]

N. D. Alikakos, Lp-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar

[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. Google Scholar

[3]

X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv:1707.09235.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jber. DMV, 105 (2003), 103-165. Google Scholar

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