-
Previous Article
On a climatological energy balance model with continents distribution
- DCDS Home
- This Issue
-
Next Article
Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions
Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem
1. | University of Oviedo, Spain, Spain, Spain |
References:
[1] |
M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biol., 23 (1985), 1-13.
doi: 10.1007/BF00276555. |
[2] |
M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2013), 131-147.
doi: 10.3934/nhm.2013.8.131. |
[3] |
M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250.
doi: 10.4171/IFB/233. |
[4] |
M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Diff. Equ. Appl., 4 (2012), 137-157.
doi: 10.7153/dea-04-09. |
[5] |
S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol., 16 (1983), 181-198.
doi: 10.1007/BF00276056. |
[6] |
M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. Med. Biol., 23 (2006), 197-229.
doi: 10.1093/imammb/dql009. |
[7] |
J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology, Arch. Ration. Mech. Anal., 194 (2009), 75-103.
doi: 10.1007/s00205-008-0164-y. |
[8] |
J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets. II. A quasilinear equation in climatology, J. Math. Anal. Appl., 352 (2009), 475-495.
doi: 10.1016/j.jmaa.2008.09.046. |
[9] |
G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118.
doi: 10.1007/s002050050146. |
[10] |
G. Galiano and V. Selgas, On a cross-diffusion segregation problem arising from a model of interacting particles, Nonlinear Anal. Real World Appl., 18 (2014), 34-49.
doi: 10.1016/j.nonrwa.2014.02.001. |
[11] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp. |
[12] |
M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94. |
[13] |
A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis. Vol. 2: Measure. The Lebesgue Integral. Hilbert Space, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm Graylock Press, Albany, N.Y. 1961. ix+128 pp. |
[14] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Quasillinear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, 1968. |
[15] |
O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968. xviii+495 pp. |
[16] |
S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 61 (2005), 257-273.
doi: 10.1007/3-7643-7317-2_19. |
[17] |
S. Shmarev, Interfaces in multidimensional diffusion equations with absorption terms, Nonlinear Anal., 53 (2003), 791-828.
doi: 10.1016/S0362-546X(03)00034-8. |
[18] |
S. Shmarev and J. L. Vazquez, The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 465-497.
doi: 10.1007/BF01193831. |
show all references
References:
[1] |
M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biol., 23 (1985), 1-13.
doi: 10.1007/BF00276555. |
[2] |
M. Bertsch, M. Mimura and T. Wakasa, Modeling contact inhibition of growth: Traveling waves, Netw. Heterog. Media, 8 (2013), 131-147.
doi: 10.3934/nhm.2013.8.131. |
[3] |
M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250.
doi: 10.4171/IFB/233. |
[4] |
M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Diff. Equ. Appl., 4 (2012), 137-157.
doi: 10.7153/dea-04-09. |
[5] |
S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol., 16 (1983), 181-198.
doi: 10.1007/BF00276056. |
[6] |
M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. Med. Biol., 23 (2006), 197-229.
doi: 10.1093/imammb/dql009. |
[7] |
J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology, Arch. Ration. Mech. Anal., 194 (2009), 75-103.
doi: 10.1007/s00205-008-0164-y. |
[8] |
J. I. Díaz and S. Shmarev, Lagrangian approach to the study of level sets. II. A quasilinear equation in climatology, J. Math. Anal. Appl., 352 (2009), 475-495.
doi: 10.1016/j.jmaa.2008.09.046. |
[9] |
G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118.
doi: 10.1007/s002050050146. |
[10] |
G. Galiano and V. Selgas, On a cross-diffusion segregation problem arising from a model of interacting particles, Nonlinear Anal. Real World Appl., 18 (2014), 34-49.
doi: 10.1016/j.nonrwa.2014.02.001. |
[11] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp. |
[12] |
M. E. Gurtin and A. C. Pipkin, On interacting populations that disperse to avoid crowding, Q. Appl. Math., 42 (1984), 87-94. |
[13] |
A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis. Vol. 2: Measure. The Lebesgue Integral. Hilbert Space, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm Graylock Press, Albany, N.Y. 1961. ix+128 pp. |
[14] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Quasillinear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, 1968. |
[15] |
O. Ladyzhenskaya and N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968. xviii+495 pp. |
[16] |
S. Shmarev, Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 61 (2005), 257-273.
doi: 10.1007/3-7643-7317-2_19. |
[17] |
S. Shmarev, Interfaces in multidimensional diffusion equations with absorption terms, Nonlinear Anal., 53 (2003), 791-828.
doi: 10.1016/S0362-546X(03)00034-8. |
[18] |
S. Shmarev and J. L. Vazquez, The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 465-497.
doi: 10.1007/BF01193831. |
[1] |
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 |
[2] |
Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185 |
[3] |
Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193 |
[4] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 |
[5] |
Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725 |
[6] |
Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 |
[7] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 |
[8] |
F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239 |
[9] |
Mostafa Bendahmane, Kenneth H. Karlsen. Martingale solutions of stochastic nonlocal cross-diffusion systems. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022024 |
[10] |
Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145 |
[11] |
Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i |
[12] |
Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010 |
[13] |
Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179 |
[14] |
Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 |
[15] |
Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211 |
[16] |
Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152 |
[17] |
Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719 |
[18] |
Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193 |
[19] |
Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145 |
[20] |
Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]