# American Institute of Mathematical Sciences

October  2011, 16(3): 739-766. doi: 10.3934/dcdsb.2011.16.739

## Existence of radial stationary solutions for a system in combustion theory

 1 INRA, Equipe BIOSP, Centre de Recherche d'Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France 2 Departamento de Ingeniería Matemática and CMM (UMI 2807 CNRS), Universidad de Chile, Blanco Encalada 2120 - 5 Piso, Santiago, Chile

Received  June 2010 Revised  February 2011 Published  June 2011

In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.
Citation: Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739
##### References:
 [1] R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,", Clarendon Press, (1975). Google Scholar [2] J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer,, SIAM J. Appl. Math., 67 (2006), 116. doi: 10.1137/050636516. Google Scholar [3] H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées,, Nonlinear Partial Differential Equations and Their Applications, (1991), 1987. Google Scholar [4] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory,, Dynamical Issues in Combustion Theory (Minneapolis, (1989), 37. Google Scholar [5] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits,, SIAM J. Math. Anal., 16 (1985), 1207. doi: 10.1137/0516088. Google Scholar [7] H. Bockhorn, J. Fröhlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls,, Combustion Theory and Modelling, 3 (1999), 177. doi: 10.1088/1364-7830/3/1/010. Google Scholar [8] I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs,, Combustion and Flame, 110 (1997), 524. doi: 10.1016/S0010-2180(97)00001-1. Google Scholar [9] J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions,, Proc. Roy. Soc. London Ser. A, 430 (1990), 479. doi: 10.1098/rspa.1990.0102. Google Scholar [10] J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls,, Combustion and Flame, 79 (1990), 381. doi: 10.1016/0010-2180(90)90147-J. Google Scholar [11] J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses,, Combustion and Flame, 84 (1991), 411. doi: 10.1016/0010-2180(91)90015-4. Google Scholar [12] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi: 10.1137/0151066. Google Scholar [13] R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar [14] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97. Google Scholar [15] Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi: 10.1006/jdeq.2000.3932. Google Scholar [16] V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits,, SIAM J. Math. Anal., 21 (1990), 1305. doi: 10.1137/0521072. Google Scholar [17] L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses,, Preprint Publications du Laboratoire d'Analyse Numérique R95032, (1995). Google Scholar [18] V. Guyonne and L. Lorenzi, Instability in a flame ball problem,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315. Google Scholar [19] V. Guyonne and P. Noble, On a model of flame ball with radiative transfer,, SIAM J. Appl. Math., 67 (2007), 854. doi: 10.1137/060659612. Google Scholar [20] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion,, Dynamical Systems and Applications, (1995), 343. Google Scholar [21] L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames,, Combust. Flame, 108 (1997), 220. doi: 10.1016/S0010-2180(96)00108-3. Google Scholar [22] L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls,, Math. Comput. Simulation, 65 (2004), 511. doi: 10.1016/j.matcom.2004.01.013. Google Scholar [23] A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry,, SIAM J. Appl. Math., 38 (1980), 382. doi: 10.1137/0138032. Google Scholar [24] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar [25] C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames,, Ann. Mat. Pura Appl., 183 (2004), 173. doi: 10.1007/s10231-003-0085-1. Google Scholar [26] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi: 10.1137/0151066. Google Scholar [27] B. Lewis and G. von Elbe, "Combustion, Flames and Explosion of Gases,", 3rd ed., (1987). Google Scholar [28] M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature,, Nonlinear Anal., 9 (1985), 1269. doi: 10.1016/0362-546X(85)90035-5. Google Scholar [29] S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls,, Combust. Theory Model., 5 (2001), 609. doi: 10.1088/1364-7830/5/4/306. Google Scholar [30] P. Ronney, Near-Limit Flame Structures at Low Lewis Number,, Combustion and Flame, 82 (1990), 1. doi: 10.1016/0010-2180(90)90074-2. Google Scholar [31] L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions,, European J. Appl. Math., 16 (2005), 741. doi: 10.1017/S0956792505006431. Google Scholar [32] G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation,, Ann. Mat. Pura Appl., 185 (2006), 273. Google Scholar [33] A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium,, Combustion Theory and Modelling, 4 (2000), 511. doi: 10.1088/1364-7830/4/4/308. Google Scholar [34] Y. B. Zeldovich, A theory of the limit of slow flame propagation,, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159. Google Scholar

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##### References:
 [1] R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,", Clarendon Press, (1975). Google Scholar [2] J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer,, SIAM J. Appl. Math., 67 (2006), 116. doi: 10.1137/050636516. Google Scholar [3] H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées,, Nonlinear Partial Differential Equations and Their Applications, (1991), 1987. Google Scholar [4] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory,, Dynamical Issues in Combustion Theory (Minneapolis, (1989), 37. Google Scholar [5] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits,, SIAM J. Math. Anal., 16 (1985), 1207. doi: 10.1137/0516088. Google Scholar [7] H. Bockhorn, J. Fröhlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls,, Combustion Theory and Modelling, 3 (1999), 177. doi: 10.1088/1364-7830/3/1/010. Google Scholar [8] I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs,, Combustion and Flame, 110 (1997), 524. doi: 10.1016/S0010-2180(97)00001-1. Google Scholar [9] J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions,, Proc. Roy. Soc. London Ser. A, 430 (1990), 479. doi: 10.1098/rspa.1990.0102. Google Scholar [10] J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls,, Combustion and Flame, 79 (1990), 381. doi: 10.1016/0010-2180(90)90147-J. Google Scholar [11] J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses,, Combustion and Flame, 84 (1991), 411. doi: 10.1016/0010-2180(91)90015-4. Google Scholar [12] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi: 10.1137/0151066. Google Scholar [13] R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar [14] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97. Google Scholar [15] Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi: 10.1006/jdeq.2000.3932. Google Scholar [16] V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits,, SIAM J. Math. Anal., 21 (1990), 1305. doi: 10.1137/0521072. Google Scholar [17] L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses,, Preprint Publications du Laboratoire d'Analyse Numérique R95032, (1995). Google Scholar [18] V. Guyonne and L. Lorenzi, Instability in a flame ball problem,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315. Google Scholar [19] V. Guyonne and P. Noble, On a model of flame ball with radiative transfer,, SIAM J. Appl. Math., 67 (2007), 854. doi: 10.1137/060659612. Google Scholar [20] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion,, Dynamical Systems and Applications, (1995), 343. Google Scholar [21] L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames,, Combust. Flame, 108 (1997), 220. doi: 10.1016/S0010-2180(96)00108-3. Google Scholar [22] L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls,, Math. Comput. Simulation, 65 (2004), 511. doi: 10.1016/j.matcom.2004.01.013. Google Scholar [23] A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry,, SIAM J. Appl. Math., 38 (1980), 382. doi: 10.1137/0138032. Google Scholar [24] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar [25] C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames,, Ann. Mat. Pura Appl., 183 (2004), 173. doi: 10.1007/s10231-003-0085-1. Google Scholar [26] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi: 10.1137/0151066. Google Scholar [27] B. Lewis and G. von Elbe, "Combustion, Flames and Explosion of Gases,", 3rd ed., (1987). Google Scholar [28] M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature,, Nonlinear Anal., 9 (1985), 1269. doi: 10.1016/0362-546X(85)90035-5. Google Scholar [29] S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls,, Combust. Theory Model., 5 (2001), 609. doi: 10.1088/1364-7830/5/4/306. Google Scholar [30] P. Ronney, Near-Limit Flame Structures at Low Lewis Number,, Combustion and Flame, 82 (1990), 1. doi: 10.1016/0010-2180(90)90074-2. Google Scholar [31] L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions,, European J. Appl. Math., 16 (2005), 741. doi: 10.1017/S0956792505006431. Google Scholar [32] G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation,, Ann. Mat. Pura Appl., 185 (2006), 273. Google Scholar [33] A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium,, Combustion Theory and Modelling, 4 (2000), 511. doi: 10.1088/1364-7830/4/4/308. Google Scholar [34] Y. B. Zeldovich, A theory of the limit of slow flame propagation,, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159. Google Scholar
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