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

Cristelle Barillon; Georgy M. Makhviladze; Vitaly A. Volpert

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 3, page 555-573
  • ISSN: 0764-583X

Abstract

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

How to cite

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Barillon, Cristelle, Makhviladze, Georgy M., and Volpert, Vitaly A.. "existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 555-573. <http://eudml.org/doc/197489>.

@article{Barillon2010,
abstract = { The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function. },
author = {Barillon, Cristelle, Makhviladze, Georgy M., Volpert, Vitaly A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Elliptic - algebraic equations; heat explosion; classical and critical solutions; topological degree; continuous branches of solutions; minimax representation.; classical solution; critical solution; multivalued solution; two-phase medium; star-shaped domain; reacting particles; elliptic-algebraic system of equations; conditions for existence of solutions},
language = {eng},
month = {3},
number = {3},
pages = {555-573},
publisher = {EDP Sciences},
title = {existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium},
url = {http://eudml.org/doc/197489},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Barillon, Cristelle
AU - Makhviladze, Georgy M.
AU - Volpert, Vitaly A.
TI - existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 555
EP - 573
AB - The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.
LA - eng
KW - Elliptic - algebraic equations; heat explosion; classical and critical solutions; topological degree; continuous branches of solutions; minimax representation.; classical solution; critical solution; multivalued solution; two-phase medium; star-shaped domain; reacting particles; elliptic-algebraic system of equations; conditions for existence of solutions
UR - http://eudml.org/doc/197489
ER -

References

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  1. 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.  
  2. 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.  
  3. W.H. Fleming and R. Rishel, An integral formula for total gradient variation. Arch. Math.11 (1960) 218-222.  Zbl0094.26301
  4. T. Gallouet, F. Mignot and J.-P. Puel, Quelques résultats sur le problème -Δu = λeu. C. R. Acad. Sci. Paris Sér. I307 (1988) 289-292.  
  5. Y. Giga, R. Kohn, Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math. XLII (1989) 845-884.  Zbl0703.35020
  6. 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.  
  7. N.V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, AMS, Graduate studies in Mathematics (1996).  Zbl0865.35001
  8. 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.  
  9. N. Mizogushi and T. Suzuki, Equations of gas combustion: S-shaped bifurcations and mushrooms. J. Differential Equations134 (1997) 183-205.  
  10. R.E. O'Malley and L.V. Kalachev, Regularization of nonlinear differential-algebraic equations. SIAM J. Math. Anal.25 (1994) 615-629.  Zbl0794.34004
  11. E.N. Rumanov and B.I. Khaikin, Critical autoignition conditions for a system of particles. Combustion, Explosion, and Shock Waves 5 (1969) 129-136.  
  12. A.I. Volpert, The spaces BV and quasilinear equations. Math USSR - Sbornik2 (1967) 225-267.  
  13. A.I. Volpert, S. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff Publishers, Dordrecht (1985).  
  14. 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).  

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