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

Refined regularity and stabilization properties in a degenerate haptotaxis system

Michael Winkler 1,, and  Christian Stinner 2,

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) $
.
Keywords: Haptotaxis, degenerate diffusion, refined regularity, large time behavior.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35D30, 35K65, 92C17.
Citation: Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[4]

S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature, 213 (1967), 256-260.  doi: 10.1038/213256a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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