Decay and boundedness results for a model of laminar flames with complex chemistry

Author:
Joel D. Avrin

Journal:
Proc. Amer. Math. Soc. **110** (1990), 989-995

MSC:
Primary 80A25; Secondary 35K57, 80A30

DOI:
https://doi.org/10.1090/S0002-9939-1990-1028038-8

MathSciNet review:
1028038

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the reaction-diffusion equations modeling two-step reactions with Arrhenius kinetics on bounded spatial domains or over all of . After noting the existence, uniqueness, and nonnegativity of global strong solutions with virtually arbitrary nonnegative initial data, we give conditions on the initial temperature that guarantee decay of the concentrations to zero and a supremum norm bound on the temperature. In our first such result we assume that the initial temperature is uniformly bounded above the two ignition temperatures. Specializing to the case of bounded spatial domains, we replace this condition by the more general requirement that the average of over the domain is above both ignition temperatures. Finally, we note a boundedness result with equal diffusion coefficients that holds for arbitrary choices of the other parameters. Combining this assumption with the hypotheses, noted above, about the initial temperature, we obtain steady-state convergence results for the temperature as well as the concentrations.

**[1]**Joel D. Avrin,*Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data*, J. Differential Equations**84**(1990), no. 2, 290–308. MR**1047571**, https://doi.org/10.1016/0022-0396(90)90080-9**[2]**-,*Qualitative theory of the Cauchy problem for a one-step reaction model on bounded domains*, submitted.**[3]**Henri Berestycki, Basil Nicolaenko, and Bruno Scheurer,*Traveling wave solutions to combustion models and their singular limits*, SIAM J. Math. Anal.**16**(1985), no. 6, 1207–1242. MR**807905**, https://doi.org/10.1137/0516088**[4]**J. D. Buckmaster and G. S. S. Ludford,*Theory of laminar flames*, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge-New York, 1982. Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, 2. MR**666866****[5]**P. Clavin,*Dynamical behavior of premixed fronts in laminar and turbulent flows*, Progr. Energy Comb. Sci.**11**(1985), 1-59.**[6]**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861****[7]**G. I. Sivashinsky,*Instabilities, pattern formation, and turbulence in flames*, Ann. Rev. Fluid Mech.**15**(1988), 179-199.**[8]**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****[9]**D. Terman,*Connection problems arising from nonlinear diffusion equations*, Proc. Microconference on Nonlinear Diffusion (J. Serrin, L. Peletier, W.-M. Ni, eds.), Berkeley, California, 1986.**[10]**-,*An application of the Cauchy index to combustion*, Dynamics of Infinite Dimensional Systems, NATO AS1 series, vol. F37 (S.-N. Chow, J. K. Hale, eds.) Springer-Verlag, Berlin and Heidelberg, 1987.**[11]**-,*Stability of planar wave solutions to a combustion model*, preprint.**[12]**F. Williams,*Combustion theory*, 2nd ed., Addison-Wesley, Reading, MA, 1985.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
80A25,
35K57,
80A30

Retrieve articles in all journals with MSC: 80A25, 35K57, 80A30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1028038-8

Article copyright:
© Copyright 1990
American Mathematical Society