Blow-up in difference schemes that approximate Neumann problems for nonlinear parabolic equations
Abstract
In this paper, based on the discrete analogue of comparison theorems and Jensen's inequality, blow-up conditions and upper bound of blow-up time of the solution of implicit finite-different problem which approximates Neumann problems for various nonlinear parabolic equations are obtained. Blow-up conditions and upper bound of blow-up time of approximated differential problems are given, which are obtained and based on comparison theorems and Jensen's inequality
About the Author
D. A. Schadinskii
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
References
1. Fujita H. On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α. J. Fac. Sci., Univ. Tokyo, Sect. I., 1966, vol. 13, pp. 109–124.
2. Levine H. A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = −Au + F(u). Arch. Ration. Mech. Anal., 1973, vol. 51, pp. 371–386.
3. Nakagawa T. Blowing up of a finite difference solution to ut = uxx + u2. Appl. Math. Optim., 1976, vol. 2, pp. 337–350.
4. Chen Y. Asymptotic behaviours of blowing-up solutions for finite difference analogue of ut = uxx + u1+α. J. Fac. Sci., Univ. Tokyo, Sect. I A., 1986, vol. 33, pp. 541–574.
5. Hartman P. Ordinary differential equations. New York, London, Sydney, John Wiley and Sons, Inc. 1964, 612 p.
6. Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P. Blow-Up in Quasilinear Parabolic Equations. Berlin, New York, De Gruyter, 1995, 554 p.
7. Matus P. P., Paradzinska A., Schadinsky D. A. Discrete analogs of the theorems comparison and their applications. Doklady of the National Academy of Sciences of Belarus, 2013, vol. 57, no. 4, pp. 16–20. (in Russian).
Review
For citations:
Schadinskii D.A. Blow-up in difference schemes that approximate Neumann problems for nonlinear parabolic equations. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(2):43-55. (In Russ.)
Views: 8
Email this article 