# American Institute of Mathematical Sciences

June  2020, 40(6): 4039-4058. doi: 10.3934/dcds.2020030

## Refined regularity and stabilization properties in a degenerate haptotaxis system

 1 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany 2 Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

* Corresponding author: Michael Winkler

Received  January 2019 Revised  May 2019 Published  October 2019

We consider the degenerate haptotaxis system
 $\left\{ \begin{array}{l} u_t = (d(x)u)_{xx} - (d(x)uw_x )_x, \\ w_t = -ug(w), \end{array} \right.$
endowed with no-flux boundary conditions in a bounded open interval
 $\Omega \subset \mathbb{R}$
. It was proposed as a basic model for haptotactic migration in heterogeneous environments. If the diffusion is degenerate in the sense that
 $d$
is non-negative, has a non-empty zero set and satisfies
 $\int_\Omega \frac{1}{d} <\infty$
, then it has been shown in [12] under appropriate assumptions on the initial data that the system has a global generalized solution satisfying in particular
 $u(\cdot,t) \rightharpoonup \frac{\mu_\infty}{d}$
weakly in
 $L^1 (\Omega)$
as
 $t \to \infty$
for some positive constant
 $\mu_\infty$
.
We now prove that under the additional restriction
 $\int_\Omega \frac{1}{d^2} <\infty$
we have the strong convergence
 $u(\cdot,t)\to \frac{\mu_\infty}{d}$
in
 $L^p (\Omega)$
as
 $t \to \infty$
for any
 $p \in (1,2)$
. In addition, with the same restriction on
 $d$
we obtain improved regularity properties of
 $u$
, for instance
 $du \in L^\infty ((0,\infty); L^p(\Omega))$
for any
 $p \in (1,\infty)$
.
Citation: Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
##### 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] J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theoret. Biol., 334 (2013), 1-12.  doi: 10.1016/j.jtbi.2013.05.033. [3] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5. [4] S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0. [5] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399. [6] 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. [7] 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. [8] K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Cont. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151. [9] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [10] C. Surulescu and M. Winkler, Global weak solutions to a strongly degenerate haptotaxis model, Commun. Math. Sci., 15 (2017), 1581-1616.  doi: 10.4310/CMS.2017.v15.n6.a5. [11] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [12] M. Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl., 112 (2018), 118-169.  doi: 10.1016/j.matpur.2017.11.002. [13] A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29 pp. doi: 10.1007/s00033-016-0741-0.

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##### 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] J. Belmonte-Beitia, T. E. Woolley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, J. Theoret. Biol., 334 (2013), 1-12.  doi: 10.1016/j.jtbi.2013.05.033. [3] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5. [4] S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0. [5] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399. [6] 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. [7] 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. [8] K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Cont. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151. [9] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [10] C. Surulescu and M. Winkler, Global weak solutions to a strongly degenerate haptotaxis model, Commun. Math. Sci., 15 (2017), 1581-1616.  doi: 10.4310/CMS.2017.v15.n6.a5. [11] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [12] M. Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl., 112 (2018), 118-169.  doi: 10.1016/j.matpur.2017.11.002. [13] A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29 pp. doi: 10.1007/s00033-016-0741-0.
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