Herrero, Miguel A. and Velázquez, J.J. L.
(1993)
*Plane structures in thermal runaway.*
Israel Journal of Mathematics , 81
(3).
pp. 321-341.
ISSN 0021-2172

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Official URL: http://www.springerlink.com/content/g078m201p243232v/

## Abstract

We consider the problem (1) u(t) = u(xx) + e(u) when x is-an-element-of R, t > 0, (2) u (x, 0) = u0(x) when x is-an-element-of R, where u0(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite time T, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile of u(x, T) near its blow-up points, which are shown to be isolated.

Item Type: | Article |
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Uncontrolled Keywords: | Semilinear heat-equations; point blow-up |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17326 |

References: | S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96. S. B. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988), 545-568. A. Bressan, On the asymptotic shape of blow-up, Indiana Univ. Math. J. 39 (1990), 947-960. A. Bressan, Stable blow-up patterns, J. Diff. Equations 98 (1992), 57-75. J. Bebernes and S. Bricher, Final time blow-up profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal. 23 (1992), 852-869. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied. Math. Sciences 83, Springer, Berlin, 1989. J. Bebernes, A. Bressan and D. Eberly, A description of blow-up for the solid fuel ignition model, Indiana Univ. Math. J. 36 (1987), 131-136. L. A. Caffarelli and A. Friedman, Blow-up of solutions of semilinear heat equations, J. Math. Anal. Appl. 129 (1988), 409-419. X. Y. Chen and H. Matano, Convergence, asymptotic periodicity and finite point blow-up in one-dimensional semilinear heat equations, J. Diff. Equations 78 (1989), 160-190. J. W. Dold, Analysis of the early stage of thermal runaway, Quart. J. Mech. Appl. Math. 38 (1985), 361-387. S. Filippas and R. V. Kohn, Refined asymptotics for the blow-up of ut - Delta(u) = uP, Comm. Pure Appl. Math. XLV (1992), 821-869. A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447. Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure. Appl. Math. 38 (1985), 297-319. Y. Giga and R. V. Kohn Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure. Appl. Math. 42 (1989), 845-884 V. A. Galaktionov and S. A. Posashkov, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Diff. Equations 22 (1986), 1165-1173. V. A. Galaktionov, M. A. Herrero and J. J. L. Velázquez, The space structure near a blow-up point for semilinear heat equations: a formal approach, USSR Comput. Math. and Math. Physics 31 (1991), 399-411. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. M. A. Herrero and J. J. L. Velázquez, Blow-up behaviour of one dimensional semilinear parabolic equations, Ann. Inst. Henri Poincaré, to appear. M. A. Herrero and J. J. L. Velázquez, Flat blow-up in one-dimensional semilinear heat equations, Differential and Integral Equations 5 (1992), 973-997. M. A. Herrero and J. J. L. Velázquez, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. in PDE 17 (1992), 205-219. F. B. Weissler, Single-point blow-up of semilinear initial value problems, J. Diff. Equations 55 (1984), 204-224. |

Deposited On: | 05 Dec 2012 09:30 |

Last Modified: | 07 Feb 2014 09:45 |

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