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
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.  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.  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.  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.  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.  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, (2003), 153.   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.  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.  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.  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.  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.  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., (2010), 147.  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.  doi: 10.1016/S0377-0427(00)00571-9.  Google Scholar

[14]

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

[15]

W. Liu, The blow-up rate of solutions of semilinear heat equation,, J. Differential Equations, 77 (1989), 104.  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.  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, 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.  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.   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.  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, (1995).   Google Scholar

[23]

F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations, 55 (1985), 204.  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.  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.  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.  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.  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.  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.  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, (2003), 153.   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.  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.  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.  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.  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.  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., (2010), 147.  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.  doi: 10.1016/S0377-0427(00)00571-9.  Google Scholar

[14]

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

[15]

W. Liu, The blow-up rate of solutions of semilinear heat equation,, J. Differential Equations, 77 (1989), 104.  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.  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, 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.  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.   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.  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, (1995).   Google Scholar

[23]

F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations, 55 (1985), 204.  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.  doi: 10.1016/S0362-546X(01)00849-5.  Google Scholar

[1]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[2]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[3]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[4]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[5]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[6]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[9]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[10]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[11]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[14]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[15]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[16]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[17]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[18]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[19]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

[Back to Top]