Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

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January 2017, 37(1): 627-643. doi: 10.3934/dcds.2017026

Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

Jiashan Zheng ^,

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

* Corresponding author: zhengjiashan2008@163.com

Received January 2016 Revised September 2016 Published November 2016

This paper deals with the Neumann problem for the coupled quasilinear chemotaxis-haptotaxis model of cancer invasion given by

$\left\{ \begin{gathered} ut = \nabla \cdot \left( {{{\left( {u + 1} \right)}^{m - 1}}\nabla u} \right) - \nabla \cdot \left( {u\nabla v} \right) - \nabla \cdot \left( {u\nabla w} \right) + u\left( {1 - u - w} \right), \hfill \\ ut = \Delta v - v + u, \hfill \\ wt = - vw, \hfill \\ \end{gathered} \right.$

where the parameter $m≥q1$ and $\mathbb{R}^N(N≥q2)$ is a bounded domain with smooth boundary. If $m>\frac{2N}{N+2}$, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results of this paper partly extend previous results of several authors.

Keywords: Boundedness, chemotaxis-haptotaxis, global existence, logistic source.

Mathematics Subject Classification: Primary:92C17, 35K55;Secondary:35K59, 35K2.

Citation: Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026

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[8]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. I., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.
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[10]	A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107. doi: 10.1142/S0218202505003915.
[11]	D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.
[12]	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, Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, B26 (2011), 159-175.
[13]	M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
[14]	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.
[15]	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.
[16]	J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), 1-33. doi: 10.1007/s00033-016-0620-8.
[17]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.
[18]	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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[20]	E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.
[21]	G. Liţanu 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.
[22]	G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.
[23]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[24]	C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321. doi: 10.1093/imamat/hxu055.
[25]	C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.
[26]	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.
[27]	Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, Mathematics, 70 (2014), 165-174.
[28]	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.
[29]	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.
[30]	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.
[31]	Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis mode, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084. doi: 10.1017/S0308210512000571.
[32]	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.
[33]	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.
[34]	Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250. doi: 10.1137/15M1014115.
[35]	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.
[36]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[37]	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.
[38]	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.
[39]	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.
[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]	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.

show all references

References:

[1]	N. D. Alikakos, $L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646. doi: 10.1142/S0218202508002796.
[3]	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.
[4]	X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11-13 pp. doi: 10.1007/s00033-015-0601-3.
[5]	M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, L. Preziosi, ed., Chapman Hall/CRC, Boca Raton, FL, (2003), 269-297.
[6]	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., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947.
[7]	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.
[8]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. I., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.
[9]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[10]	A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107. doi: 10.1142/S0218202505003915.
[11]	D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.
[12]	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, Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, B26 (2011), 159-175.
[13]	M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
[14]	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.
[15]	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.
[16]	J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), 1-33. doi: 10.1007/s00033-016-0620-8.
[17]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.
[18]	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.
[19]	E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[20]	E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.
[21]	G. Liţanu 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.
[22]	G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213. doi: 10.3934/dcdsb.2015.20.189.
[23]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[24]	C. Stinner, C. Surulescu and G. Meral, A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321. doi: 10.1093/imamat/hxu055.
[25]	C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.
[26]	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.
[27]	Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, Mathematics, 70 (2014), 165-174.
[28]	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.
[29]	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.
[30]	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.
[31]	Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis mode, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084. doi: 10.1017/S0308210512000571.
[32]	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.
[33]	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.
[34]	Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250. doi: 10.1137/15M1014115.
[35]	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.
[36]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[37]	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.
[38]	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.
[39]	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.
[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]	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.

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