References
L. Abia, J.C. López-Marcos and J. Martínez, Blow-up for semidiscretizations of reaction-diffusion equations. Appl. Numer. Math.,20 (1996), 145–156.
L. Abia, J.C. López-Marcos and J. Martínez, On the blow-up time convergence of semi-discretizations of reaction-diffusion equations. Appl. Numer. Math.,26 (1998), 399–414.
L.M. Abia, J.C. López-Marcos and J. Martínez, The Euler method in the numerical integration of reaction-diffusion problems with blow-up. Appl. Numer. Math.,38 (2001), 287–313.
L.A. Caffarrelli and A. Friedman, Blow-up of solutions of nonlinear heat equations. J. Math. Anal. Appl.,129 (1988), 409–419.
C.-H. Cho, On a finite difference scheme for the parabolic blow-up problems. Master’s thesis, Research Institute for Mathematical Sciecnes, Kyoto University, 2005.
Y.-G. Chen, Asymptotic behaviours of blowing-up solutions for finite difference analogue ofu t =u xx +u 1+α. J. Fac. Sci., Univ. Tokyo,33 (1986), 541–574.
P.J. Davis and P. Rabinowitz, Methods of Numerical Integration. Academis Press, 1975.
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J.,34 (1985), 425–447.
C. Hirota and K. Ozawa, A methof of estimating the blow-up time and blow-up rate of the solution of the system of ordinary sifferencial equations—An application to the blow-up problems of partial differencial equations—(in Japanese). Trans. Japan SIAM,14 (2004), 13–38.
S. Itô, On blow-up of positive solutions of semilinear parabolic equations. J. Fac. Sci. Univ. Tokyo, Sect. IA,37 (1990), 527–536.
A.A. Lacey, Global blow-up of a nonlinear heat equation. Proc. R. Soc. Edin.,104 A (1986), 161–167.
T. Nakagawa, Blowing up of a finite difference solution tou t = uxx +u 2. Appl. Math. Optim.,2 (1976), 337–350.
T. Nakagawa and T. Ushijima, Finite element analysis of the semi-linear heat equation of blow-up type. in Topics in Numer. Anal. III, ed. J.J.H. Miller, 1977, 275–291.
W. Ren and X.-P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions. J. Comp. Phys.,159 (2000), 246–273.
M. Tabata, A finite difference approach to the number of peaks of solutions for semilinear parabolic problems. J. Math. Soc. Japan,32 (1980), 171–191.
T.K. Ushijima, On the approximation of blow-up time for solutions of nonlinear parabolic equations. Publ. RIMS,36 (2000), 613–640.
F. Weissler, Single point blowup of semilinear initial value problems. J. Diff. Eqns.,55 (1984), 202–224.
Author information
Authors and Affiliations
About this article
Cite this article
Cho, C.H., Hamada, S. & Okamoto, H. On the finite difference approximation for a parabolic blow-up problem. Japan J. Indust. Appl. Math. 24, 131–160 (2007). https://doi.org/10.1007/BF03167529
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167529