May  2016, 21(3): 849-861. doi: 10.3934/dcdsb.2016.21.849

Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory

1. 

Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada

2. 

School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China

Received  September 2014 Revised  November 2015 Published  January 2016

Uniqueness of nonzero positive solutions of a Laplacian elliptic equation arising in combustion theory is of great interest in combustion theory since it can be applied to determine where the extinction phenomenon occurs. We study the uniqueness whenever the orders of the reaction rates are in $(-\infty,1]$. Previous results on uniqueness treated the case when the orders belong to $[0,1)$. When the orders are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, but there is little study on uniqueness. Our results on the uniqueness are completely new when the orders are negative or 1, and also improve some known results when the orders belong to $(0,1)$. Our results provide exact intervals of the Frank-Kamenetskii parameters on which the extinction phenomenon never occurs. The novelty of our methodology is to combine and utilize the results from Laplacian elliptic inequalities and equations to derive new results on uniqueness of nonzero positive solutions for general Laplacian elliptic equations.
Citation: Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 849-861. doi: 10.3934/dcdsb.2016.21.849
References:
[1]

I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782. doi: 10.1090/S0002-9939-1993-1116249-5.

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary problems, Indiana Univ. Math. J., 71 (1972), 125-146. doi: 10.1512/iumj.1972.21.21012.

[3]

T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. Roy. Soc. London, Sect. A, 368 (1979), 441-461.

[4]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8.

[5]

K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1.

[6]

N. P. Cac, On the uniqueness of positive solutions of a nonlinear elliptic boundary value problems, J. London Math. Soc., 25 (1982), 347-354. doi: 10.1112/jlms/s2-25.2.347.

[7]

Y. H. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32 (2000), 707-733. doi: 10.1137/S0036141098343586.

[8]

Y. H. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1977), 209-243. doi: 10.1007/BF01221125.

[10]

P. Hess, On the uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18. doi: 10.1007/BF01215108.

[11]

W. P. Ho, R. B. Barat and J. W. Bozzelli, Thermal reaction of CH2C12 in H2/02 mixtures: Implications for chlorine inhibition of CO conversion to CO2, Combust. Flame, 88 (1992), 265-295.

[12]

K. Q. Lan, Nonzero positive solutions of systems of elliptic boundary value problems, Proc. Amer. Math. Soc., 139 (2011), 4343-4349. doi: 10.1090/S0002-9939-2011-10840-2.

[13]

K. Q. Lan, A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities, J. Differential Equations, 246 (2009), 909-928. doi: 10.1016/j.jde.2008.10.007.

[14]

K. Q. Lan, Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities, J. Math. Anal. Appl., 380 (2011), 520-530. doi: 10.1016/j.jmaa.2011.03.030.

[15]

K. Q. Lan, A fixed point theory for weakly inward S-contractive maps, Nonlinear Anal., 45 (2001), 189-201. doi: 10.1016/S0362-546X(99)00337-5.

[16]

K. Q. Lan and W. Lin, A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities, Nonlinear Anal., 74 (2011), 5415-5425. doi: 10.1016/j.na.2011.05.025.

[17]

K. Q. Lan and J. R. L. Webb, Variational inequalities and fixed point theorems for PM-maps, J. Math. Anal. Appl., 224 (1998), 102-116. doi: 10.1006/jmaa.1998.5988.

[18]

K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475.

[19]

K. Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl., 394 (2012), 581-591. doi: 10.1016/j.jmaa.2012.04.061.

[20]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.

[21]

G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame, Combust. Flame, 101 (1995), 101-112.

[22]

W. M. Ni, Uniqueness of solutions of nonlinear Dirichelet problems, J. Differential Equations, 50 (1983), 289-304. doi: 10.1016/0022-0396(83)90079-7.

[23]

W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108. doi: 10.1002/cpa.3160380105.

[24]

S. S. Okoya, Boundness for a system of reaction-diffusion equations. I, Mathematika, 41 (1994), 293-300. doi: 10.1112/S0025579300007397.

[25]

J. A. Smoller and A. G. Wasserman, Existence, uniqueness, and non degeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys., 95 (1984), 129-159. doi: 10.1007/BF01468138.

[26]

K. Taira, Semilinear elliptic boundary-value problems in combustion theory, Proc. Roy. Soc. Edinburgh, Sect. A, 132 (2002), 1453-1476.

[27]

D. G. Vlachos, The interplay of transport, kinetics, and thermal interactions in the stability of premixed hydrogen/air flames, Combust. Flame, 103 (1995), 59-75. doi: 10.1016/0010-2180(95)00072-E.

[28]

G. C. Wake, T. Boddington and P. Gray, Thermal explosion and the disappearance of criticality in systems with distribution temperatures, IV. Rigonus bounds and their practical relevance, Proc. Roy. Soc. London, Sect. A, 425 (1989), 285-289.

[29]

S. H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X.

[30]

S. H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. Roy. Soc. London, Sect. A 454 (1998), 1031-1048. doi: 10.1098/rspa.1998.0195.

[31]

F. A. Williams, Combustion theory, 2nd ed, Redwood City, CA: Addison-Wesley, (1985), 585-588.

show all references

References:
[1]

I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782. doi: 10.1090/S0002-9939-1993-1116249-5.

[2]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary problems, Indiana Univ. Math. J., 71 (1972), 125-146. doi: 10.1512/iumj.1972.21.21012.

[3]

T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: Exact results for the slab, Proc. Roy. Soc. London, Sect. A, 368 (1979), 441-461.

[4]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55-64. doi: 10.1016/0362-546X(86)90011-8.

[5]

K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1.

[6]

N. P. Cac, On the uniqueness of positive solutions of a nonlinear elliptic boundary value problems, J. London Math. Soc., 25 (1982), 347-354. doi: 10.1112/jlms/s2-25.2.347.

[7]

Y. H. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32 (2000), 707-733. doi: 10.1137/S0036141098343586.

[8]

Y. H. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1977), 209-243. doi: 10.1007/BF01221125.

[10]

P. Hess, On the uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18. doi: 10.1007/BF01215108.

[11]

W. P. Ho, R. B. Barat and J. W. Bozzelli, Thermal reaction of CH2C12 in H2/02 mixtures: Implications for chlorine inhibition of CO conversion to CO2, Combust. Flame, 88 (1992), 265-295.

[12]

K. Q. Lan, Nonzero positive solutions of systems of elliptic boundary value problems, Proc. Amer. Math. Soc., 139 (2011), 4343-4349. doi: 10.1090/S0002-9939-2011-10840-2.

[13]

K. Q. Lan, A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities, J. Differential Equations, 246 (2009), 909-928. doi: 10.1016/j.jde.2008.10.007.

[14]

K. Q. Lan, Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities, J. Math. Anal. Appl., 380 (2011), 520-530. doi: 10.1016/j.jmaa.2011.03.030.

[15]

K. Q. Lan, A fixed point theory for weakly inward S-contractive maps, Nonlinear Anal., 45 (2001), 189-201. doi: 10.1016/S0362-546X(99)00337-5.

[16]

K. Q. Lan and W. Lin, A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities, Nonlinear Anal., 74 (2011), 5415-5425. doi: 10.1016/j.na.2011.05.025.

[17]

K. Q. Lan and J. R. L. Webb, Variational inequalities and fixed point theorems for PM-maps, J. Math. Anal. Appl., 224 (1998), 102-116. doi: 10.1006/jmaa.1998.5988.

[18]

K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421. doi: 10.1006/jdeq.1998.3475.

[19]

K. Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of p-Laplace equations, J. Math. Anal. Appl., 394 (2012), 581-591. doi: 10.1016/j.jmaa.2012.04.061.

[20]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.

[21]

G. P. Miller, The structure of a stoichiometric CCI4-CH4-air flat flame, Combust. Flame, 101 (1995), 101-112.

[22]

W. M. Ni, Uniqueness of solutions of nonlinear Dirichelet problems, J. Differential Equations, 50 (1983), 289-304. doi: 10.1016/0022-0396(83)90079-7.

[23]

W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108. doi: 10.1002/cpa.3160380105.

[24]

S. S. Okoya, Boundness for a system of reaction-diffusion equations. I, Mathematika, 41 (1994), 293-300. doi: 10.1112/S0025579300007397.

[25]

J. A. Smoller and A. G. Wasserman, Existence, uniqueness, and non degeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys., 95 (1984), 129-159. doi: 10.1007/BF01468138.

[26]

K. Taira, Semilinear elliptic boundary-value problems in combustion theory, Proc. Roy. Soc. Edinburgh, Sect. A, 132 (2002), 1453-1476.

[27]

D. G. Vlachos, The interplay of transport, kinetics, and thermal interactions in the stability of premixed hydrogen/air flames, Combust. Flame, 103 (1995), 59-75. doi: 10.1016/0010-2180(95)00072-E.

[28]

G. C. Wake, T. Boddington and P. Gray, Thermal explosion and the disappearance of criticality in systems with distribution temperatures, IV. Rigonus bounds and their practical relevance, Proc. Roy. Soc. London, Sect. A, 425 (1989), 285-289.

[29]

S. H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X.

[30]

S. H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. Roy. Soc. London, Sect. A 454 (1998), 1031-1048. doi: 10.1098/rspa.1998.0195.

[31]

F. A. Williams, Combustion theory, 2nd ed, Redwood City, CA: Addison-Wesley, (1985), 585-588.

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