American Institute of Mathematical Sciences

Advanced
January  2007, 7(1): 125-144. doi: 10.3934/dcdsb.2007.7.125

Global existence for chemotaxis with finite sampling radius

T. Hillen 1, , K. Painter 2, and  Christian Schmeiser 3,

1. 

University of Alberta, Edmonton, Alberta, Canada T6G 2G1, Canada
2. 

Heriot-Watt University, Edinburgh, EH11 1UF, United Kingdom
3. 

University of Vienna, Faculty for Mathematics, Nordbergstraße 15, 1090 Wien, Austria

Received  January 2006 Revised  September 2006 Published  October 2006

Migrating cells measure the external environment through receptor-binding of specific chemicals at their outer cell membrane. In this paper this non-local sampling is incorporated into a chemotactic model. The existence of global bounded solutions of the non-local model is proven for bounded and unbounded domains in any space dimension. According to a recent classification of spikes and plateaus, it is shown that steady state solutions cannot be of spike-type. Finally, numerical simulations support the theoretical results, illustrating the ability of the model to give rise to pattern formation. Some biologically relevant extensions of the model are also considered.
Keywords: Finite Sampling Radius, Non-local Gradient., Chemotaxis, Global Existence.
Mathematics Subject Classification: Primary: 92C17 Secondary: 35K1.
Citation: T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[1]

Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
[2]

Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[3]

Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475
[4]

Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2515-2535. doi: 10.3934/dcdsb.2021146
[5]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[6]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032
[7]

Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193
[8]

Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks and Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107
[9]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[10]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[11]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002
[12]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[13]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[14]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[15]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
[16]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[17]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations and Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036
[18]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[19]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[20]

Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (187)
  • HTML views (0)
  • Cited by (45)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy