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A double saddle-node bifurcation theorem
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 |
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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