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July  2019, 18(4): 2047-2067. doi: 10.3934/cpaa.2019092

A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production

1. 

College of Science, Donghua University, Shanghai 200051, China

2. 

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Received  September 2018 Revised  November 2018 Published  January 2019

Fund Project: Y. Tao acknowledges support of the National Natural Science Foundation of China (No. 11571070). M. Winkler was supported by the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks

We consider the chemotaxis-haptotaxis system
$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), \\ v_t = \Delta v-v+f(u), \\ w_t = -vw+\eta w(1-u-w), \end{array} \right. \end{eqnarray*} $
in a bounded convex domain
$ \Omega\subset \mathbb{R} ^n $
with smooth boundary, where
$ \chi, \xi, \mu $
and
$ \eta $
are positive constants, and where
$ f \in C^1([0,\infty)) $
is a given function fulfilling
$ f(0) \ge 0 $
and
$ \begin{eqnarray*} f(s) \le K_f (s+1)^\alpha \qquad \mbox{for all } s\ge 0 \end{eqnarray*} $
with some
$ K_f >0 $
and
$ \alpha>0 $
.
It is asserted that whenever
$ \begin{eqnarray*} \alpha < \left\{ \begin{array}{ll} \frac{3}{2} \qquad & \mbox{if } n = 1, \\ \frac{n+6}{2(n+2)} \qquad & \mbox{if } n\ge 2, \end{array} \right. \end{eqnarray*} $
the Neumann boundary problem with suitably regular initial data possesses a unique global and bounded classical solution.
Citation: 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
References:
[1]

N. Bellomo, A. Bellouquid, 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 system, Z. Angew. Math. Phys., 67 (2016), 11. doi: 10.1007/s00033-015-0601-3.  Google Scholar

[3]

Z. Chen and Y. Tao, Large-data solution in a three-dimensional chemotaxis-haptotaxis system with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Appl Math, (2018). doi: 10.1007/s10440-018-0216-8.  Google Scholar

[4]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 18 (2005), 1685–1734. doi: 10.1142/S0218202505000947.  Google Scholar

[5]

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

[6]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330–1355. doi: 10.1137/S0036141001385046.  Google Scholar

[7]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138–163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

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Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94. doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

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[10]

T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165–198. doi: 10.1142/S0218202512500480.  Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Eq., 215 (2005), 52–107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23 (1968), Providence, RI.  Google Scholar

[13]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564–1595. doi: 10.1088/0951-7715/29/5/1564.  Google Scholar

[14]

P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Rat. Mech. Anal., 74 (1980), 335–353. doi: 10.1007/BF00249679.  Google Scholar

[15]

P. Y. H. Pang and Y. Wang, Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Eq., 263 (2017), 1269–1292. doi: 10.1016/j.jde.2017.03.016.  Google Scholar

[16]

P. Y. H. Pang and Y. Wang, Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Mod. Meth. Appl. Sci., 28 (2018), 2211–2235. doi: 10.1142/S0218202518400134.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. Google Scholar

[20]

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

[21]

Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Proc. Roy. Soc. Edinb. Sect A, 144 (2014), 1067–1084. doi: 10.1017/S0308210512000571.  Google Scholar

[22]

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

[23]

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. Differential Eq., 257 (2014), 784–815. doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[24]

Y. Tao and M. Winkler, Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250. doi: 10.1137/15M1014115.  Google Scholar

[25]

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

[26]

L. Wang, C. Mu, X. Hu and Y. Tian, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Math. Meth. Appl. Sci., 40 (2017), 3000–3016. doi: 10.1002/mma.4216.  Google Scholar

[27]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Differential Equations, 260 (2016), 6960–6988. doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[28]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Eq., 248 (2010), 2889–2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[29]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Eq., 264 (2018), 6109–6151. doi: 10.1016/j.jde.2018.01.027.  Google Scholar

[30]

M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, Preprint. Google Scholar

[31]

J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discr. Cont. Dyn. Syst., 37 (2017), 627–643. doi: 10.3934/dcds.2017026.  Google Scholar

show all references

References:
[1]

N. Bellomo, A. Bellouquid, 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 system, Z. Angew. Math. Phys., 67 (2016), 11. doi: 10.1007/s00033-015-0601-3.  Google Scholar

[3]

Z. Chen and Y. Tao, Large-data solution in a three-dimensional chemotaxis-haptotaxis system with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Appl Math, (2018). doi: 10.1007/s10440-018-0216-8.  Google Scholar

[4]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 18 (2005), 1685–1734. doi: 10.1142/S0218202505000947.  Google Scholar

[5]

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

[6]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330–1355. doi: 10.1137/S0036141001385046.  Google Scholar

[7]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138–163. doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[8]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94. doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[9]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647–1669. doi: 10.1080/03605309708821314.  Google Scholar

[10]

T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165–198. doi: 10.1142/S0218202512500480.  Google Scholar

[11]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Eq., 215 (2005), 52–107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc. Transl., 23 (1968), Providence, RI.  Google Scholar

[13]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564–1595. doi: 10.1088/0951-7715/29/5/1564.  Google Scholar

[14]

P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Rat. Mech. Anal., 74 (1980), 335–353. doi: 10.1007/BF00249679.  Google Scholar

[15]

P. Y. H. Pang and Y. Wang, Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Eq., 263 (2017), 1269–1292. doi: 10.1016/j.jde.2017.03.016.  Google Scholar

[16]

P. Y. H. Pang and Y. Wang, Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Mod. Meth. Appl. Sci., 28 (2018), 2211–2235. doi: 10.1142/S0218202518400134.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. Google Scholar

[20]

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

[21]

Y. Tao and M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Proc. Roy. Soc. Edinb. Sect A, 144 (2014), 1067–1084. doi: 10.1017/S0308210512000571.  Google Scholar

[22]

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

[23]

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. Differential Eq., 257 (2014), 784–815. doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[24]

Y. Tao and M. Winkler, Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250. doi: 10.1137/15M1014115.  Google Scholar

[25]

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

[26]

L. Wang, C. Mu, X. Hu and Y. Tian, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Math. Meth. Appl. Sci., 40 (2017), 3000–3016. doi: 10.1002/mma.4216.  Google Scholar

[27]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Differential Equations, 260 (2016), 6960–6988. doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[28]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Eq., 248 (2010), 2889–2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[29]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Eq., 264 (2018), 6109–6151. doi: 10.1016/j.jde.2018.01.027.  Google Scholar

[30]

M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, Preprint. Google Scholar

[31]

J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discr. Cont. Dyn. Syst., 37 (2017), 627–643. doi: 10.3934/dcds.2017026.  Google Scholar

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