# American Institute of Mathematical Sciences

November  2013, 12(6): 2935-2946. doi: 10.3934/cpaa.2013.12.2935

## Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition

 1 College of Mathematics and statistics, Chongqing University, Chongqing 401331, China, China, China, China

Received  November 2011 Revised  March 2012 Published  May 2013

This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point $x=0$ for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.
Citation: Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
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##### References:
 [1] Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010 [2] Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 [3] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [4] Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733 [5] Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001 [6] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [7] C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 [8] Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443 [9] Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155 [10] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [11] W. Edward Olmstead, Colleen M. Kirk, Catherine A. Roberts. Blow-up in a subdiffusive medium with advection. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1655-1667. doi: 10.3934/dcds.2010.28.1655 [12] Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691 [13] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [14] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [15] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [16] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [17] Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 [18] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [19] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [20] Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

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