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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2020, 40 (2) : 907-932. doi: 10.3934/dcds.2020066

[4]

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

[5]

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 & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055

[19]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65

[20]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]