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

Ling Liu; Jiashan Zheng

doi:10.3934/dcdsb.2018324

Previous Article
Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion
DCDS-B Home
This Issue
Next Article
Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation

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

Ling Liu ^1, and Jiashan Zheng ^2,3,

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:

[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[51]	J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topological Methods in Nonlinear Analysis, 49 (2017), 463-480. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[51]	J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topological Methods in Nonlinear Analysis, 49 (2017), 463-480. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[2]	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
[3]	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
[4]	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
[5]	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
[6]	Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026
[7]	Youshan Tao, Michael Winkler. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2047-2067. doi: 10.3934/cpaa.2019092
[8]	Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[9]	Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Decay in chemotaxis systems with a logistic term. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 257-268. doi: 10.3934/dcdss.2020014
[10]	Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125
[11]	Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147
[12]	Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
[13]	Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216
[14]	Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 139-164. doi: 10.3934/dcdss.2020008
[15]	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
[16]	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
[17]	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
[18]	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
[19]	Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
[20]	Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors