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

Previous Article
Retraction
DCDS-B Home
This Issue
Next Article
Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations

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].

Keywords: New criterion, chemotaxis, critical dimension, global boundedness, interpolation inequality.

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

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, L^p-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	N. Bellomo, A. Bellouquid, Y. 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 l^p-l^q 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.

show all references

References:

[1]	N. D. Alikakos, L^p-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	N. Bellomo, A. Bellouquid, Y. 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 l^p-l^q 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.

[1]	Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
[2]	Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262
[3]	Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168
[4]	Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069
[5]	Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2018328
[6]	Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[7]	Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035
[8]	Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[9]	Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[10]	Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463
[11]	Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang, Yi-An Chen. On global boundedness of the Chen system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1673-1681. doi: 10.3934/dcdsb.2017080
[12]	Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 165-176. doi: 10.3934/dcdss.2020009
[13]	Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
[14]	Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
[15]	Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132
[16]	Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[17]	Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
[18]	Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258
[19]	Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013
[20]	Gleb Beliakov. Construction of aggregation operators for automated decision making via optimal interpolation and global optimization. Journal of Industrial & Management Optimization, 2007, 3 (2) : 193-208. doi: 10.3934/jimo.2007.3.193

American Institute of Mathematical Sciences