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existence of solutions for an elliptic-algebraic systemdescribing heat explosion in a two-phase medium

Published online by Cambridge University Press:  15 April 2002

Cristelle Barillon ,
Georgy M. Makhviladze  and
Vitaly A. Volpert
Cristelle Barillon
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel.
Georgy M. Makhviladze
Affiliation:
Center for Research in Fire and Explosion Studies, University of Central Lancashire, Preston, PR1 2HE, UK.
Vitaly A. Volpert
Affiliation:
Analyse Numérique, UMR 5585 CNRS, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France.
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Abstract

The paper is devoted to analysis of an elliptic-algebraic system ofequations describing heat explosion in a two phase medium filling a star-shaped domain. Three typesof solutions are found: classical, critical andmultivalued. Regularity of solutions is studied as well as theirbehavior depending on the size of the domain and on the coefficient ofheat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

Keywords

Elliptic - algebraic equationsheat explosionclassical and critical solutionstopological degreecontinuous branchesof solutionsminimax representation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 555 - 573
Copyright
© EDP Sciences, SMAI, 2000

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References

E.V. Chernenko and V.I. Rozenband, Calculation of the extremal combustion characteristics of aerial suspensions of metal with autoignition. Combustion, Explosion and Shock Waves 16 (1980) 3-10.
I.G. Dik and A.Yu. Krainov, Ignition regims of a gas suspension in a vessel with heated walls. Combustion, Explosion, and Shock Waves 20 (1984) 58-61.
Fleming, W.H. and Rishel, R., An integral formula for total gradient variation. Arch. Math. 11 (1960) 218-222. CrossRef
Gallouet, T., Mignot, F. and Puel, J.-P., Quelques résultats sur le problème -Δu = λeu . C. R. Acad. Sci. Paris Sér. I 307 (1988) 289-292.
Y. Giga, R. Kohn, Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math. XLII (1989) 845-884.
M.A. Gurevich, G.E. Ozerova and A.M. Stepanov, Ignition limit of a monofractional gas suspension. Combustion, Explosion, and Shock Waves 10 (1974) 88-93.
N.V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, AMS, Graduate studies in Mathematics (1996).
V.I. Lisitsyn, E.N. Rumanov and B.I. Khaikin, Induction period in the ignition of a particle system. Combustion Explosion, and Shock Waves 1 (1971) 1-6.
Mizogushi, N. and Suzuki, T., Equations of gas combustion: S-shaped bifurcations and mushrooms. J. Differential Equations 134 (1997) 183-205. CrossRef
O'Malley, R.E. and Kalachev, L.V., Regularization of nonlinear differential-algebraic equations. SIAM J. Math. Anal. 25 (1994) 615-629. CrossRef
E.N. Rumanov and B.I. Khaikin, Critical autoignition conditions for a system of particles. Combustion, Explosion, and Shock Waves 5 (1969) 129-136.
Volpert, A.I., The spaces BV and quasilinear equations. Math USSR - Sbornik 2 (1967) 225-267. CrossRef
A.I. Volpert, S. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff Publishers, Dordrecht (1985).
Ya. B. Zeldovich, G.I. Barenblatt, V.B. Librovich and G.M. Makhviladze, The mathematical theory of combustion and explosion. Plenum Press, New York-London (1985).