Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

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July 2019, 24(7): 3357-3377. doi: 10.3934/dcdsb.2018324

Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

1.	Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China
2.	School of Information, Renmin University of China, Beijing, 100872, China
3.	School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received March 2018 Revised August 2018 Published January 2019

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source

$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), \\ {v_t = \Delta v- v +u}, \quad \\ {w_t = - vw}, \quad\\ {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, \quad x\in \partial\Omega, t>0, \\ {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), \quad x\in \Omega, \ \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$

in a smooth bounded domain

$ \mathbb{R}^N(N\geq1) $

, with parameter

$ r>1 $

. the parameters

$ a\in \mathbb{R}, \mu>0, \chi>0 $

. It is shown that when

$ r>2 $

, or

$ \begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \ \end{array} \end{equation*} $

the considered problem possesses a global classical solution which is bounded, where

$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $

is a positive constant which is corresponding to the maximal sobolev regularity. Here

$ C_{\beta} $

is a positive constant which depends on

$ \xi $

$ \|u_0\|_{C(\bar{\Omega})}, \ \|v_0\|_{W^{1, \infty}(\Omega)} $

and

$ \|w_0\|_{L^\infty(\Omega)} $

. This result improves or extends previous results of several authors.

Keywords: Boundedness, chemotaxis–haptotaxis, (generalized) logistic sourcen.

Mathematics Subject Classification: 92C17, 35K55, 35K59, 35K20.

Citation: 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

References:

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[2]	X. Cao, Boundedness in a three-dimensional chemotaxis–haptotaxis model, Z. Angew. Math. Phys., 67 (2015), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3.
[3]	M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 11 (2005), 1685-1734. doi: 10.1142/S0218202505000947.
[4]	M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399.
[5]	T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Diff. Eqns., 252 (2012), 5832-5851.
[6]	H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations, in: Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, Res. Inst. Math. Sci. (RIMS), Kyoto, 26 (2011), 159–175.
[7]	T. Hillen and K. J. Painter, A use's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[8]	T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480.
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[11]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
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[14]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[15]	G. Liţcanu and C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758. doi: 10.1142/S0218202510004775.
[16]	J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp. doi: 10.1007/s00033-016-0620-8.
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[19]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[20]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
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[22]	Y. Tao, Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039.
[23]	Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174.
[24]	Y. Tao and M. Wang, Global solution for a chemotactic–haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238. doi: 10.1088/0951-7715/21/10/002.
[25]	Y. Tao and M. Wang, A combined chemotaxis–haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558. doi: 10.1137/090751542.
[26]	Y. Tao and M. Winkler, A chemotaxis–haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[27]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[28]	Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084. doi: 10.1017/S0308210512000571.
[29]	Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis–haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.
[30]	Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014.
[31]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[32]	C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713. doi: 10.1137/060655122.
[33]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.
[34]	Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.
[35]	Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Diff. Eqns., 260 (2016), 6960-6988. doi: 10.1016/j.jde.2016.01.017.
[36]	M. Winkler, Does a volume-filling effect always prevent chemotactic collapse, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[37]	M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[38]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[39]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[40]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[41]	M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056. doi: 10.1088/1361-6544/aaaa0e.
[42]	M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.
[43]	M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[44]	T. Xiang, Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source, J. Diff. Eqns., 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032.
[45]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.
[46]	J. Zheng, Optimal controls of multi-dimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125. doi: 10.1080/00207179.2015.1038587.
[47]	J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[48]	J. Zheng, Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888. doi: 10.1016/j.jmaa.2015.05.071.
[49]	J. Zheng, Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear a logistic source, J. Math. Anal. Appl., 450 (2017), 1047-1061. doi: 10.1016/j.jmaa.2017.01.043.
[50]	J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi–linear chemotaxis system with logistic source, Zeitschriftfür Angewandte Mathematik und Mechanik, 97 (2017), 414–421. doi: 10.1002/zamm.201600166.
[51]	J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topological Methods in Nonlinear Analysis, 49 (2017), 463-480.
[52]	J. Zheng, Boundedness of solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source, Nonlinearity, 30 (2017), 1987-2009. doi: 10.1088/1361-6544/aa675e.
[53]	J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion, Discrete and Continuous Dynamical Systems, 37 (2017), 627-643. doi: 10.3934/dcds.2017026.
[54]	J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Diff. Eqns., 263 (2017), 2606-2629. doi: 10.1016/j.jde.2017.04.005.
[55]	J. Zheng, A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867–888, arXiv: 1712.00906, 2017. doi: 10.1016/j.jmaa.2015.05.071.
[56]	J. Zheng and Y. Wang, Boundedness of solutions to a quasilinear chemotaxis–haptotaxis model, Compu. Math. Appl., 71 (2016), 1898-1909. doi: 10.1016/j.camwa.2016.03.014.
[57]	P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Disc. Cont. Dyna. Syst., 36 (2015), 1737-1757. doi: 10.3934/dcds.2016.36.1737.

show all references

References:

[1]	N. Bellomo, A. Belloquid, 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.
[2]	X. Cao, Boundedness in a three-dimensional chemotaxis–haptotaxis model, Z. Angew. Math. Phys., 67 (2015), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3.
[3]	M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 11 (2005), 1685-1734. doi: 10.1142/S0218202505000947.
[4]	M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399.
[5]	T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Diff. Eqns., 252 (2012), 5832-5851.
[6]	H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations, in: Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, Res. Inst. Math. Sci. (RIMS), Kyoto, 26 (2011), 159–175.
[7]	T. Hillen and K. J. Painter, A use's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[8]	T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480.
[9]	D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I. Jahresberichte der Deutschen Mathematiker-Vereinigung, 105 (2003), 103-165.
[10]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[11]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
[12]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.
[13]	E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
[14]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[15]	G. Liţcanu and C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758. doi: 10.1142/S0218202510004775.
[16]	J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp. doi: 10.1007/s00033-016-0620-8.
[17]	A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476. doi: 10.1142/S0218202510004301.
[18]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105.
[19]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[20]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[21]	J. Simon, Compact sets in the space L^p(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[22]	Y. Tao, Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039.
[23]	Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174.
[24]	Y. Tao and M. Wang, Global solution for a chemotactic–haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238. doi: 10.1088/0951-7715/21/10/002.
[25]	Y. Tao and M. Wang, A combined chemotaxis–haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558. doi: 10.1137/090751542.
[26]	Y. Tao and M. Winkler, A chemotaxis–haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[27]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[28]	Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084. doi: 10.1017/S0308210512000571.
[29]	Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis–haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.
[30]	Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014.
[31]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[32]	C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713. doi: 10.1137/060655122.
[33]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.
[34]	Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.
[35]	Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Diff. Eqns., 260 (2016), 6960-6988. doi: 10.1016/j.jde.2016.01.017.
[36]	M. Winkler, Does a volume-filling effect always prevent chemotactic collapse, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[37]	M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[38]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[39]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[40]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[41]	M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056. doi: 10.1088/1361-6544/aaaa0e.
[42]	M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.
[43]	M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.
[44]	T. Xiang, Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source, J. Diff. Eqns., 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032.
[45]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.
[46]	J. Zheng, Optimal controls of multi-dimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125. doi: 10.1080/00207179.2015.1038587.
[47]	J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[48]	J. Zheng, Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888. doi: 10.1016/j.jmaa.2015.05.071.
[49]	J. Zheng, Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear a logistic source, J. Math. Anal. Appl., 450 (2017), 1047-1061. doi: 10.1016/j.jmaa.2017.01.043.
[50]	J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi–linear chemotaxis system with logistic source, Zeitschriftfür Angewandte Mathematik und Mechanik, 97 (2017), 414–421. doi: 10.1002/zamm.201600166.
[51]	J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topological Methods in Nonlinear Analysis, 49 (2017), 463-480.
[52]	J. Zheng, Boundedness of solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source, Nonlinearity, 30 (2017), 1987-2009. doi: 10.1088/1361-6544/aa675e.
[53]	J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion, Discrete and Continuous Dynamical Systems, 37 (2017), 627-643. doi: 10.3934/dcds.2017026.
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