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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. [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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