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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

Shouming Zhou 1, , Chunlai Mu 1, , Yongsheng Mi 1, and  Fuchen Zhang 1,

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.
Keywords: Nonlocal diffusion equation, blow-up, single point blow-up., blow-up rates.
Mathematics Subject Classification: 35B40, 45E1.
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 and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
References:
[1]

F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ., 8 (2008), 189-215. doi: 10.1007/s00028-007-0377-9.

[2]

M. Bogoya, R. Ferreira and J. D. Rossi, A nonlocal nonlinear diffusion equation with blowing up boundary conditions, J. Math. Anal. Appl., 337 (2008), 1284-1294. doi: 10.1016/j.jmaa.2007.04.049.

[3]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.

[4]

C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problem that approximate the heat equation with Dirichlet boundary condition, Israel J. Math., 170 (2009), 53-60. doi: 10.1007/s11856-009-0019-8.

[5]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390. doi: 10.1016/j.jde.2006.12.002.

[6]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, Springer, Berlin, (2003), 153-191.

[7]

A. Friedman and J. B. Mcleod, Blow-up of positive solution of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.

[8]

V. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dynam. Syst. A, 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.

[9]

J. Garcia-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.

[10]

J. Garcia-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004.

[11]

J. Garcia-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Comm. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.

[12]

J. Garcia-Melián and F. Quirós, Fujita exponents for evolution problems with nonlocal diffusion, J. Evol. Equ., 10 (2010), 147-161. doi: 10.1007/s00028-009-0043-5.

[13]

P. Groisman and J. D. Rossi, Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions, J. Comput. Appl. Math., 135 (2001), 135-155. doi: 10.1016/S0377-0427(00)00571-9.

[14]

L. Hopf, Introduction to differential equations of physics, Dover, New York, 8 (1948), 55-100.

[15]

W. Liu, The blow-up rate of solutions of semilinear heat equation, J. Differential Equations, 77 (1989), 104-122. doi: 10.1016/0022-0396(89)90159-9.

[16]

A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 8 (1983), 1350-1366. doi: 10.1137/0143090.

[17]

J. D. Murray, "Mathematical Biology," Springer New York, 1993.

[18]

P. Morse and H. Feshback, Methods of theoretical physics, McGraw Hill, New York, 1 (1953).

[19]

S. X. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y.

[20]

A. F. Pazoto and J. D. Rossi, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Anal., 52 (2007), 143-155.

[21]

M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640. doi: 10.1016/j.na.2008.02.076.

[22]

A. Samarski, V. A. Galaktionov, S. P. Kurdyunov and A. P. Mikailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, (1995).

[23]

F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1985), 204-224. doi: 10.1016/0022-0396(84)90081-0.

[24]

S. N. Zheng, L. Z. Zhao and F. Chen, Blow-up rates in a parabolic system of ignition model, Nonlinear Anal., 51 (2002), 663-672. doi: 10.1016/S0362-546X(01)00849-5.

show all references

References:
[1]

F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ., 8 (2008), 189-215. doi: 10.1007/s00028-007-0377-9.

[2]

M. Bogoya, R. Ferreira and J. D. Rossi, A nonlocal nonlinear diffusion equation with blowing up boundary conditions, J. Math. Anal. Appl., 337 (2008), 1284-1294. doi: 10.1016/j.jmaa.2007.04.049.

[3]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.

[4]

C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problem that approximate the heat equation with Dirichlet boundary condition, Israel J. Math., 170 (2009), 53-60. doi: 10.1007/s11856-009-0019-8.

[5]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390. doi: 10.1016/j.jde.2006.12.002.

[6]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, Springer, Berlin, (2003), 153-191.

[7]

A. Friedman and J. B. Mcleod, Blow-up of positive solution of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.

[8]

V. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dynam. Syst. A, 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.

[9]

J. Garcia-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.

[10]

J. Garcia-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004.

[11]

J. Garcia-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Comm. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.

[12]

J. Garcia-Melián and F. Quirós, Fujita exponents for evolution problems with nonlocal diffusion, J. Evol. Equ., 10 (2010), 147-161. doi: 10.1007/s00028-009-0043-5.

[13]

P. Groisman and J. D. Rossi, Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions, J. Comput. Appl. Math., 135 (2001), 135-155. doi: 10.1016/S0377-0427(00)00571-9.

[14]

L. Hopf, Introduction to differential equations of physics, Dover, New York, 8 (1948), 55-100.

[15]

W. Liu, The blow-up rate of solutions of semilinear heat equation, J. Differential Equations, 77 (1989), 104-122. doi: 10.1016/0022-0396(89)90159-9.

[16]

A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 8 (1983), 1350-1366. doi: 10.1137/0143090.

[17]

J. D. Murray, "Mathematical Biology," Springer New York, 1993.

[18]

P. Morse and H. Feshback, Methods of theoretical physics, McGraw Hill, New York, 1 (1953).

[19]

S. X. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y.

[20]

A. F. Pazoto and J. D. Rossi, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Anal., 52 (2007), 143-155.

[21]

M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640. doi: 10.1016/j.na.2008.02.076.

[22]

A. Samarski, V. A. Galaktionov, S. P. Kurdyunov and A. P. Mikailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, (1995).

[23]

F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations, 55 (1985), 204-224. doi: 10.1016/0022-0396(84)90081-0.

[24]

S. N. Zheng, L. Z. Zhao and F. Chen, Blow-up rates in a parabolic system of ignition model, Nonlinear Anal., 51 (2002), 663-672. doi: 10.1016/S0362-546X(01)00849-5.

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