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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.
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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N. BellomoA. BelloquidY. 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

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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

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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

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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

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T. HillenK. 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

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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

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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

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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

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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

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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

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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

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V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105. Google Scholar

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K. OsakiT. TsujikawaA. 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

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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 Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360. Google Scholar

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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

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Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174. Google Scholar

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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

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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

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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

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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

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show all references

References:
[1]

N. BellomoA. BelloquidY. 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. HillenK. 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. IshidaK. 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. OsakiT. TsujikawaA. 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 Lp(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. WangC. 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

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