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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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

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