February  2010, 27(1): 301-323. doi: 10.3934/dcds.2010.27.301

On a nonlocal aggregation model with nonlinear diffusion

1. 

Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242, United States, United States

Received  April 2009 Revised  December 2009 Published  February 2010

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.
Citation: Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301
[1]

Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks and Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749

[2]

Jacob Bedrossian, Nancy Rodríguez. Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1279-1309. doi: 10.3934/dcdsb.2014.19.1279

[3]

Yuming Paul Zhang. On a class of diffusion-aggregation equations. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 907-932. doi: 10.3934/dcds.2020066

[4]

Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170

[5]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[6]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[7]

Georg Hetzer, Wenxian Shen. Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : i-iii. doi: 10.3934/dcds.2015.35.4i

[8]

Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022042

[9]

Philip K. Maini, Luisa Malaguti, Cristina Marcelli, Serena Matucci. Diffusion-aggregation processes with mono-stable reaction terms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1175-1189. doi: 10.3934/dcdsb.2006.6.1175

[10]

Jan Haškovec, Dietmar Oelz. A free boundary problem for aggregation by short range sensing and differentiated diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1461-1480. doi: 10.3934/dcdsb.2015.20.1461

[11]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819

[12]

Simone Fagioli, Yahya Jaafra. Multiple patterns formation for an aggregation/diffusion predator-prey system. Networks and Heterogeneous Media, 2021, 16 (3) : 377-411. doi: 10.3934/nhm.2021010

[13]

Razvan C. Fetecau, Hui Huang, Daniel Messenger, Weiran Sun. Zero-diffusion limit for aggregation equations over bounded domains. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022078

[14]

Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683

[15]

Armel Ovono Andami. From local to nonlocal in a diffusion model. Conference Publications, 2011, 2011 (Special) : 54-60. doi: 10.3934/proc.2011.2011.54

[16]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure and Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037

[17]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks and Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014

[18]

Laurent Desvillettes, Michèle Grillot, Philippe Grillot, Simona Mancini. Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4675-4692. doi: 10.3934/dcds.2018205

[19]

Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3463-3478. doi: 10.3934/dcdsb.2016107

[20]

Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]