Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-14T06:48:20.977Z Has data issue: false hasContentIssue false
  • English
  • Français

Regional blow-up of solutions to the initial boundary value problem for ut = uδ(Δu + u)

Published online by Cambridge University Press:  14 November 2011

Masayoshi Tsutsumi  and
Tetsuya Ishiwata
Masayoshi Tsutsumi
Affiliation:
Department of Applied Physics, Waseda University, Tokyo 169, Japan
Tetsuya Ishiwata
Affiliation:
Department of Applied Physics, Waseda University, Tokyo 169, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

Non-negative solutions of the initial boundary value problem for a degenerate parabolic equation are investigated. It is shown that solutions blow up regionally in finite tine. The size of blow-up sets is determined for radially symmetric cases.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 871 - 887
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Anada, K., Fukuda, I. and Tsutsumi, M.. Local existence and uniqueness of viscosity solutions for u 1 = uδu– y∣∇u∣2 + f(t, u). Adv. Math. Sci. Appl. (to appear).Google Scholar
2Anada, K., Fukuda, I. and Tsutsumi, M.. Regional blow-up and decay of solutions to the initialboundary value problem for u 1 = uu xx–y(u x)2 + ku 2. Funkcial. Ekvac, 39 (1996) 363–87.Google Scholar
3Angenent, S.. The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390 (1988), 79–96.Google Scholar
4Angenent, S.. On the formation of singularities in the curve shortening flow. J. Differential Geom. 33 (1991), 601–33.CrossRefGoogle Scholar
5Bertsch, M., Passo, R. Dal and Ughi, M.. Discontinuous ‘viscosity’ solutions of a degenerate parabolic equation. Trans. Amer. Math. Soc. 320 (1990), 779–98.Google Scholar
6Bertsch, M., Passo, R. Dal and Ughi, M.. Nonuniqueness of solutions of a degenerate parabolic equation. Ann. Mat. Pura Appl. (IV) 161 (1992), 57–81.CrossRefGoogle Scholar
7Bertsch, M. and Ughi, M.. Positivity properties of viscosity solutions of a degenerate parabolic equations. Nonlinear Anal. 14 (1990), 571–92.CrossRefGoogle Scholar
8Friedman, A. and McLeod, J. B.. Blow-up of solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–47.CrossRefGoogle Scholar
9Friedman, A. and McLeod, J. B.. Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Rational Mech. Anal. 96 (1987), 55–80.CrossRefGoogle Scholar
10Fukuda, I., Ishii, H. and Tsutsumi, M.. Uniqueness of solutions to the Cauchy problem for u 1 = uδu– y∣∇u∣2 = 0. Differential Integral Equations 6 (1993), 1231–52.Google Scholar
11Fukuda, I. and Tsutsumi, M.. The initial-boundary value problem for a nonlinear degenerate parabolic equation. Nonlinear Anal. 16 (1991), 997–1009.CrossRefGoogle Scholar
12Gage, M. E.. On the size of the blow-up set for a quasilinear parabolic equation. Contemp. Math. 127 (1992), 41–58.CrossRefGoogle Scholar
13Gage, M. E. and Hamilton, R. S.. The heat equation shrinking convex plane curves. J. Differential Geom. 23(1986), 69–96.CrossRefGoogle Scholar
14Galaktionov, V. A.. Asymptotic behaviour of unbounded solutions of the nonlinear parabolic equation u 1 = (u σu x)x + u σ+1. Differentsial'nye Uravneniya 21 (1985), 1126–34 (in Russian).Google Scholar
15Idogawa, T. and Tsutsumi, M.. On numerical results for the linearizable generalized curve shortening equation. Nonlinear Times Digest (to appear).Google Scholar
16Qi, Y-W.. The global feature of unbounded solutions to a nonlinear parabolic equation. J. Differential Equations 99 (1992), 139–52.CrossRefGoogle Scholar
17Sacks, R. E.. Global behaviour for a class of nonlinear evolution equations. SIAM. J. Math. Anal. 16 (1985), 233–50.CrossRefGoogle Scholar
18Tsutsumi, M.. The curve shortening equation and its generalizations (Preprint).Google Scholar
19Tsutsumi, M.. Regional blow-up of solutions to the Cauchy problem for u t = u σ(u xx + u) (Preprint).Google Scholar
20Wiegner, M.. Blow-up for solutions of some degenerate parabolic equations. Differential Integral Equations 7 (1994), 1641–7.CrossRefGoogle Scholar