-
Previous Article
Retraction
- DCDS-B Home
- This Issue
-
Next Article
Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations
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 |
$\begin{align*}\begin{cases}u_t = Δ u-χ\nabla·(u\nabla v) &\text{ in }Ω×\mathbb R^+,\\v_t = Δ v-v+w &\text{ in }Ω×\mathbb R^+,\\w_t = Δ w-w+u &\text{ in }Ω×\mathbb R^+,\end{cases}\end{align*}$ |
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. |
[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 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.
|
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. |
[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 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.
|
[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, 2017, 22 (11) : 1-21. 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 |
2017 Impact Factor: 0.972
Tools
Metrics
Other articles
by authors
[Back to Top]