References

THE BLOWING-UP PHENOMENON FOR A REACTION DIFFUSION EQUATION WITH A LOCALIZED NON LINEAR SOURCE TERM AND DIRICHLET-NEUMANN BOUNDARY CONDITIONS


[1] L. M. Abia, J. C. López-Marcos and J. Martinez, On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Applied Numerical Mathematics 26(4) (1998), 399-414.
DOI: https://doi.org/10.1016/S0168-9274(97)00105-0

[2] L. M. Abia, J. C. López-Marcos and J. Martínez, Blow-up for semidiscretizations of reaction-diffusion equations, Applied Numerical Mathematics 20(1-2) (1996), 145-156.
DOI: https://doi.org/10.1016/0168-9274(95)00122-0

[3] G. Acosta, J. Fernandez Bonder, P. Groisman and J. D. Rossi, Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions, M2AN Mathematical Modelling and Numerical Analysis 36(1) (2002), 55-68.
DOI: https://doi.org/10.1051/m2an:2002003

[4] G. Acosta, J. Fernandez Bonder, P. Groisman and J. D. Rossi, Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions, Discrete & Continuous Dynamical Systems: B (2)2 (2002), 279-294.
DOI: https://doi.org/10.3934/dcdsb.2002.2.279

[5] G. Acosta, R. G. Duran and J. D. Rossi, An adaptive time step procedure for a parabolic problem with blow-up, Computing 68(4) (2002), 343-373.
DOI: https://doi.org/10.1007/s00607-002-1449-x

[6] C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, Journal of Computational and Applied Mathematics 97(1-2) (1998), 3-22.
DOI: https://doi.org/10.1016/S0377-0427(98)00100-9

[7] M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing up solutions, Communications on Pure and Applied Mathematics 41(6) (1988), 841-863.
DOI: https://doi.org/10.1002/cpa.3160410606

[8] K. Bimpong-Beta, P. Ortoleva and J. Ross, Far-form-equilibrium phenomenon at local sites of reactions, Journal of Chemical Physics 60(8) (1974), 3124-3133.
DOI: https://doi.org/10.1063/1.1681498

[9] T. K. Boni, On the blow-up and the asymptotic behaviour of the solution of one semilinear parabolic equation of second order, Comptes Rendus de l'Académie des Sciences, Series I: Mathematics 326(3) (1998), 317-322.
DOI: https://doi.org/10.1016/S0764-4442(97)82987-4

[10] T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, Comptes Rendus de l’Académie des Sciences, Series I: Mathematics 333(8) (2001), 795-800.
DOI: https://doi.org/10.1016/S0764-4442(01)02078-X

[11] T. K. Boni, On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions, Commentationes Mathematicae Universitatis Carolinae 40(3) (1999), 457-475.

[12] C. Brandle, P. Groisman and J. D. Rossi, Fully discrete adaptive methods for a blow-up problem, Mathematical Models and Methods in Applied Sciences 14(10) (2004), 1425-1450.
DOI: https://doi.org/10.1142/S0218202504003751

[13] C. Brandle, F. Quiros and J. D. Rossi, An adaptive numerical method to handle blow-up in a parabolic system, Numerische Mathematik 102(1) (2005), 39-59.
DOI: https://doi.org/10.1007/s00211-005-0638-x

[14] H. Brezis, Analyse Fonctionnelle, Théorie et Application, Collection Mathematiques Appliquées Pour la Maitrise, Masson, Paris.

[15] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for revisited, Advances in Differential Equations 1(1) (1996), 73-90.

[16] C. J. Budd, W. Huang and R. D. Russel, Moving mesh methods for problems with blow-up, SIAM Journal on Scientific Computing 17(2) (1996), 305-327.
DOI: https://doi.org/10.1137/S1064827594272025

[17] J. M. Chadam and H. M. Yin, A diffusion equation with localized chemical reactions, Proceedings of the Edinburgh Mathematical Society 37(1) (1994), 101-118.
DOI: https://doi.org/10.1017/S0013091500018721

[18] Y. G. Chen, Asymptotic behaviours of blowing up solutions for finite difference analogue of J. Fac. Sci. Univ. Tokyo, Sec. IA, Math. 33 (1986), 541-574.

[19] A. De. Pablo, M. P. LLanos and R. Ferreira, Numerical blow-up for p-Laplacian equation with a nonlinear source, Proceedings of Equatdiff. 11 (2005), 363-367.

[20] R. G. Duran, J. I. Etcheverry and J. D. Rossi, Numerical approximation of a parabolic problem with a nonlinear boundary condition, Discrete & Continuous Dynamical Systems: A 4(3) (1998), 497-506.
DOI: https://doi.org/10.3934/dcds.1998.4.497

[21] C. M. Elliot and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM Journal on Numerical Analysis 30(6) (1993), 1622-1663.
DOI: https://doi.org/10.1137/0730084

[22] J. Fernandez Bonder and J. D. Rossi, Blow-up vs. spurious steady solutions, Proceedings of the American Mathematical Society 129(1) (2001), 139-144.
DOI: https://doi.org/10.1090/S0002-9939-00-05663-X

[23] J. Fernandez Bonder, P. Groisman and J. D. Rossi, On numerical blow-up sets, Proceedings of the American Mathematical Society 130(7) (2002), 2049-2055.
DOI: https://doi.org/10.1090/S0002-9939-02-06350-5

[24] R. Ferreira, P. Groisman and J. D. Rossi, Numerical blow-up for the porous medium equation with a source, Numerical Methods for Partial Differential Equations 20(4) (2004), 552-575.
DOI: https://doi.org/10.1002/num.10103

[25] V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete & Continuous Dynamical Systems: A 8(2) (2002), 399-433.
DOI: https://doi.org/10.3934/dcds.2002.8.399

[26] P. Groisman, Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions, Computing 76(3-4) (2006), 325-352.
DOI: https://doi.org/10.1007/s00607-005-0136-0

[27] P. Groisman and J. D. Rossi, Dependance of the blow-up time with respect to parameters and numerical approximations for a parabolic problem, Asymptotic Analysis 37(1) (2004), 79-91.

[28] P. Groisman and J. D. Rossi, Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions, Journal of Computational and Applied Mathematics 135(1) (2001), 135-155.
DOI: https://doi.org/10.1016/S0377-0427(00)00571-9

[29] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23 AMS, Providence, RI, 1968.

[30] M. N. Le Roux, Semidiscretization in time of nonlinear parabolic equations with blow-up of the solution, SIAM Journal on Numerical Analysis 131(1) (1994), 170-195.
DOI: https://doi.org/10.1137/0731009

[31] M. N. Le Roux, Semidiscretization in time of a fast diffusion equation, Journal of Mathematical Analysis and Applications 137(2) (1989), 354-370.
DOI: https://doi.org/10.1016/0022-247X(89)90251-5

[32] T. Imai and K. Mochizuki, On blow-up of solutions for quasilinear degeneate parabolic equations, Publications of the Research Institute for Mathematical Sciences 27(5) (1991), 695-709.
DOI: https://doi.org/10.2977/prims/1195169267

[33] T. Nakagawa, Blowing up on the finite difference solution to Applied Mathematics and Optimization 2(4) (1975), 337-350.
DOI: https://doi.org/10.1007/BF01448176

[34] T. K. Boni and Halima Nachid, Blow-up for semidiscretizations of some semilinear parabolic equations with nonlinear boundary conditions, Rev. Ivoir. Sci. Tech. 11 (2008), 61-70.

[35] T. K. Boni, Halima Nachid and Nabongo Diabate, Blow-up for discretization of a localized semilinear heat equation, Analele Stiintifice Ale Univertatii 2 (2010).

[36] Halima Nachid, Quenching for semi discretizations of a semilinear heat equation with potentiel and general non linearities, Revue d’Analyse Numerique et de Theorie de l’Approximation 2 (2011), 164-181.

[37] Halima Nachid, Full discretizations of solution for a semilinear heat equation with Neumann boundary condition, Research and Communications in Mathematics and Mathematical Sciences 1(1) (2012), 53-85.

[38] Halima Nachid, Behavior of the numerical quenching time with a potential and general nonlinearities, Journal of Mathematical Sciences: Advances and Application 15(2) (2012), 81-105.

[39] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1967.

[40] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Series: Birkhauser Advanced Tests/ Basler Lehrbcher, 2007.

[41] A. M. Stuart and M. S. Floater, On the computation of blow-up, European Journal of Applied Mathematics 1(1) (1990), 47-71.
DOI: https://doi.org/10.1017/S095679250000005X

[42] A. A. Samarski, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995.

[43] R. Suzuki, On blow-up sets and asymptotic behavior of interface of one dimensional quasilinear degenerate parabolic equation, Publications of the Research Institute for Mathematical Sciences 27 (1991), 375-398.

[44] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, Journal of Differential Equations 153(2) (1999), 374-406.
DOI: https://doi.org/10.1006/jdeq.1998.3535

[45] P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM Journal on Mathematical Analysis 29(6) (1998), 1301-1334.
DOI: https://doi.org/10.1137/S0036141097318900

[46] L. Wang and Q. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equation, Journal of Mathematical Analysis and Applications 200(2) (1996), 315-321.
DOI: https://doi.org/10.1006/jmaa.1996.0207

[47] W. Walter, Differential-und Integral-ungleichungen, Springer, Berlin, 1964.

[48] F. B. Weissler, An blow-up estimate for a nonlinear heat equation, Communications on Pure and Applied Mathematics 38(3) (1985), 291-295.
DOI: https://doi.org/10.1002/cpa.3160380303