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A probabilistic approach to blow-up of a semilinear heat equation

Published online by Cambridge University Press:  14 November 2011

Alison M. Etheridge
Alison M. Etheridge
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland, U.K.
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Abstract

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1235 - 1245
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Bebernes, J. and Eberly, D.. Mathematical Problems from Combustion Theory (Berlin: Springer, 1989).CrossRefGoogle Scholar
2Bramson, M. D.. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978), 531–82.CrossRefGoogle Scholar
3Champneys, A., Harris, S., Toland, J., Warren, J. and Williams, D.. Algebra, analysis and probabilityfor a system of coupled reaction-diffusion equations. Philos. Trans. Roy. Soc. London Ser. A 350 (1995), 69–112.Google Scholar
4Dawson, D. A.. The critical measure diffusion process. Z. Warsch. Verw. Gebiete 40 (1977), 125–45.CrossRefGoogle Scholar
5Dawson, D. A.. Measure-valued Markov processes. In Ecole d'été de probabilités de Saint Flour (1991), Lecture Notes in Mathematics 1541 (Berlin: Springer, 1993).CrossRefGoogle Scholar
6Donnelly, P. J. and Kurtz, T. G.. A particle representation of infinite population genetic models (preprint).Google Scholar
7Dynkin, E. B.. An Introduction to Branching Measure-valued Processes, CRM Monograph Series 6 (Providence, RI: American Mathematical Society, 1994).CrossRefGoogle Scholar
8Etheridge, A. M.. Asymptotic behaviour of measure-valued critical branching processes. Proc. Amer. Math. Soc. 118 (1993), 1251–61.CrossRefGoogle Scholar
9Etheridge, A. M.. Superprocesses, brancing processes and an explosive pde. In Israel Mathematical Conference Proceedings 10: Stochastic Analysis: Random Fields and Measure-Valued Processes, eds Fouque, J.-P., Hochberg, K. J. and Merzbach, E. (Providence, RI: American Mathematical Society, 1995).Google Scholar
10Fujita, H.. On the blowing up of solutions of the Cauchy problem for ut = Δu + u 1+∝. J. Fac. Sci. Univ. Tokyo Sect. I 13 (1996), 109–24.Google Scholar
11Ikeda, N., Nagasawa, M. and Watanabe, S.. Branching Markov processes I, II, III. J. Math. Kyoto Univ. 8 (1968), 233–78; 9 (1969), 96–160.Google Scholar
12Lacey, A. A. and Tzanetis, D.. Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition. IMA J. Appl. Math. 41 (1988), 207–15.CrossRefGoogle Scholar
13Gall, J. F. Le. Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 (1991), 1399–439.CrossRefGoogle Scholar
14Gall, J. F.Le. A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 (1993), 25–46.CrossRefGoogle Scholar
15Merle, F.. Solution of a non-linear heat equation with arbitrary give blowup points. Comm. Pure Appl. Math. 45 (1992), 263–300.CrossRefGoogle Scholar
16Skorohod, V. V.. Branching diffusion processes (translated by Seckler, B.). Theory Probab. Appl. 9 (1964), 445–9.CrossRefGoogle Scholar