Abstract
This paper studies a finite difference approximation to the similinear heat equation (1) with special emphasis on the case when the exact solution blows up with the blowing-up timeT ∞. The key results will be given in Propositions 1 and 2. Proposition 1 states the local convergence, i.e., the convergence of the proposed finite difference solution to the exact solution in any fixed time interval 0 ⩽t ⩽ T, whereT < T ∞. Proposition 2 states the convergence of the numerical blowing-up time to the exact oneT ∞.
Similar content being viewed by others
References
H. Fujita, On the blowing up of solutions to the Cauchy problem foru t = Δu+u 1+α,J. Faculty Science, Univ. of Tokyo,13 (1966), 109–124.
—, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations,Proc. Symposium in Pure Math. 18,AMS (1970), 105–113.
K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations,Proc. Japan Acad. 49 (1973), 503–505.
S. Ito, On the blowing up of solutions of semilinear parabolic equations,Sugaku 18 (1966), 44–47. (In Japanese)
S. Kaplan, On the growth of solutions of quasilinear parabolic equations,Comm. Pure Appl. Math. 16 (1963), 305–330.
W. E. Kastenberg, Space dependent reactor kinetics with positive feedback,Nukleonik 11 (1968), 126–130.
K. Kobayashi, H. Tanaka, andT. Sirao, On the growing up problem for semilinear parabolic differential equations, (to appear).
M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations,Publ. RIMS, Kyoto Univ. 8 (1972), 211–229.
—, Existence and nonexistence of global solutions of the first boundary value problem for a certain quasilinear parabolic equation,Funkcialaj Ekvacioj 17 (1974), 13–24.
Author information
Authors and Affiliations
Additional information
Communicated by M. Yamaguti
Rights and permissions
About this article
Cite this article
Nakagawa, T. Blowing up of a finite difference solution tou t = uxx + u2 . Appl Math Optim 2, 337–350 (1975). https://doi.org/10.1007/BF01448176
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01448176