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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

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