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Monotone and nonmonotone clines with partial panmixia across a geographical barrier
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 |
$ \left\{ \begin{array}{l} u_t = (d(x)u)_{xx} - (d(x)uw_x )_x, \\ w_t = -ug(w), \end{array} \right. $ |
$ \Omega \subset \mathbb{R} $ |
$ d $ |
$ \int_\Omega \frac{1}{d} <\infty $ |
$ u(\cdot,t) \rightharpoonup \frac{\mu_\infty}{d} $ |
$ L^1 (\Omega) $ |
$ t \to \infty $ |
$ \mu_\infty $ |
$ \int_\Omega \frac{1}{d^2} <\infty $ |
$ u(\cdot,t)\to \frac{\mu_\infty}{d} $ |
$ L^p (\Omega) $ |
$ t \to \infty $ |
$ p \in (1,2) $ |
$ d $ |
$ u $ |
$ du \in L^\infty ((0,\infty); L^p(\Omega)) $ |
$ p \in (1,\infty) $ |
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. |
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] |
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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