Filippas, Stathis and Herrero, Miguel A. and Velázquez, J.J. L.
(2000)
*Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity.*
Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, 456
.
pp. 2957-2982.
ISSN 1364-5021

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Official URL: http://rspa.royalsocietypublishing.org/content/456/2004/2957.full.pdf+html

## Abstract

We consider the semilinear heat equation with critical power nonlinearity. Using formal. arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign-changing solutions, which blow-up at x = 0 and t = T < ∞, for space dimension N = 3,4,5,6. These solutions exhibit fast blow-up; i.e. they satisfy lim(t up arrowT)(T - t)(1/(p-1))u(0, t) = ∞. In contrast, radial solutions that are positive and decreasing behave as in the subcritical case for any N ≥ 3. This last result is extended in the case of exponential nonlinearity and N = 2.

Item Type: | Article |
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Uncontrolled Keywords: | Matched asymptotic expansions; semilinear heat equation; blow-up; critical exponents; singularities; dynamics |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 16657 |

References: | Angenent, S. 1988 The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96. Angenent, S. & Velázquez, J. J. L. 1995 Asymptotic shape of cusp singularities in curve shortening. Duke Math. Jl 77,71–100. Bernoff, A., Bertozzi, A. & Witelski, T. 1998 Axisymmetric surface diffusion dynamics and stability of self-similar pinchoff. J. Statist. Phys. 93, 725–776. Blythe, P. A. & Crighton, D. G. 1989 Shock generated ignition: the induction zone. Proc. R. Soc. Lond. A426, 189–209. Dold, J. W. 1985 Analysis of the early stage of thermal runaway. Q. JlMe ch. Appl. Math. 38, 361–387. Filippas, S. & Kohn, R. V. 1992 Refined asymptotics for the blow-up of ut−Δu = up. Commun. Pure Appl. Math. 45, 821–869. Eggers, J. 1997 Nonlinear dynamics and breakup of free surface flows. Rev. Mod. Phys. 69,865–929. Gage, M. & Hamilton, R. S. 1987 The heat equation shrinking convex plane curves. J. Diff. Geom. 26, 285–314. Galaktionov, V. A. & Posashkov, S. A. 1986 Application of new comparison theorems in the ivestigation of unbounded solutions of nonlinear parabolic equations. Diff. Eqns 22, 1165-1173. Giga, Y. & Kohn R. V. 1985 Asymptotically self similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38, 297–319. Giga, Y. & Kohn, R. V. 1987 Characterising blow-up using similarity variables. Indiana Univ.Math. Jl 36, 1–40. Giga, Y. & Kohn, R. V. 1989 Nondegeneracy of blow-up for semilinear heat equations. Commun. Pure Appl. Math. 42, 845–884. Gidas, B. & Spruck, J. 1981 Global and local behaviour of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598. Herrero, M. A. & Velázquez, J. J. L. 1994 Explosion des solutions d’equations paraboliques semilinéaires. C. R. Acad. Sci. Paris 319, 141–145. Herrero, M. A. & Velázquez, J. J. L. 1996 Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623. Herrero, M. A. & Velázquez, J. J. L. 1997 On the melting of ice balls. SIAM JlMath. Analysis 28, 1–32. Keller, H. B. & Segel, L. A. 1970 Initiation of slime mold aggregation viewed as an instability.J. Theor. Biol. 26, 399–415. Szego, G. 1978 Orthogonal polynomials, vol. 23. Providence, RI: American Mathematical Society. Velázquez, J. J. L. 1992 Higher dimensional blow-up for semilinear parabolic equations. Commun. PDE 17, 1567–1596. Velázquez, J. J. L. 1993 Classification of singularities for blowing up solutions in higher dimensions.Trans. Am. Math. Soc. 338, 441–464. |

Deposited On: | 09 Oct 2012 09:59 |

Last Modified: | 07 Feb 2014 09:33 |

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