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]	Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020396
[2]	Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154
[3]	Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021025
[4]	Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
[5]	Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[6]	Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291
[7]	Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021011
[8]	Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[9]	Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017
[10]	Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[11]	Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 443-481. doi: 10.3934/dcdsb.2020284
[12]	Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[13]	Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020346
[14]	Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262
[15]	Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040
[16]	Yuyuan Ouyang, Trevor Squires. Some worst-case datasets of deterministic first-order methods for solving binary logistic regression. Inverse Problems & Imaging, 2021, 15 (1) : 63-77. doi: 10.3934/ipi.2020047
[17]	Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021002
[18]	Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020171
[19]	Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021002
[20]	Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286

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