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On the existence and uniqueness of solutions to a singular nonlinear boundary value problem arising in isothermal autocatalytic chemical kinetics

Published online by Cambridge University Press:  20 January 2009

D. J. Needham  and
A. C. King
D. J. Needham
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ
A. C. King
Affiliation:
Department of Mathematics, University of Keele, Keele, Staffordshire ST5 5BG
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Abstract

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In this paper we consider the questions of existence and uniqueness of solutions to a singular, nonlinear boundary value problem arising from a model problem in isothermal autocatalytical chemical kinetics. The boundary value problem occurs in the construction of a small time asymptotic solution to an initial-boundary value problem (King and Needham [14]), and existence and uniqueness for the boundary value problem are required for consistency of this formal asymptotic solution.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 479 - 500
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Bandle, C. and Levine, H. A., On the existence and non-existence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc. 665 (1989), 595–624.CrossRefGoogle Scholar
2.Bandle, C. and Stakgold, I., The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc. 286 (1984), 275–293.CrossRefGoogle Scholar
3.Berestycki, H., Lions, P. L. and Peletier, L. A., An ODE approach to the existence of positive solutions in for semilinear problems in ℝN, Indiana Univ. Math. J. 30 (1981), 141–157.CrossRefGoogle Scholar
4.Billingham, J. and Needham, D. J., A note on the properties of travelling wave solutions arising in cubic autocatalysis, Dynamic Stability Systems 6 (1991), 33–49.CrossRefGoogle Scholar
5.Billingham, J. and Needham, D. J., The development of travelling waves in quadratic and cubic autocatalysis with unequal diffussion rates I. Permanent form travelling waves, Philos. Trans. Roy. Soc. London Ser A 334 (1991), 1–24.Google Scholar
6.Billingham, J. and Needham, D. J., The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates II. An initial value problem with an immobilized or nearly immobilized autocatalyst, Philos. Trans. Roy. Soc. London Ser A 336 (1991), 4967–5539.Google Scholar
7.Billingham, J. and Needham, D. J., The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates III. Large time development in quadratic autocatalysis, Quart. Appl. Math. 50 (1992), 343–372.CrossRefGoogle Scholar
8.Coddington, E. A. and Levinson, N., The theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
9.Escobedo, M. and Zua, E. Zua, Self-similar solutions for a convection-diffusion equation with absorption in ℝN, Israel J. Math. 74 (1991), 47–64.CrossRefGoogle Scholar
10.Fujita, H., On the blowing up of solutions of the Cauchy problem for u t = Δu + u 1+α, J. Fac. Sci. Univ. Tokyo Sect. A Math. 16 (1966), 105–113.Google Scholar
11.Gray, P., Merkin, J. H., Needham, D. J. and Scott, S. K., The development of travelling waves in a simple isothermal chemical system III. Cubic and mixed autocatalysis, Proc. Roy. Soc. London Ser. A 430 (1990), 509–524.Google Scholar
12.Grundy, R. E. and Peletier, L. A., Short time behaviour of a singulation solution to the heat equation with absorption, Proc. Roy. Soc. Edinburgh 107A (1987), 271–288.CrossRefGoogle Scholar
13.Hirsch, M. W. and Smale, S., Differential equations, dynamical systems and linear algebra (Academic Press, New York, 1974).Google Scholar
14.King, A. C. and Needham, D. J., On a singular initial boundary value problem for a reaction-diffusion equation arising from a simple model of isothermal chemical autocatalysis, Proc. Roy. Soc. London Ser A 437 (1992), 657–671.Google Scholar
15.Levine, H. A., The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), 262–288.CrossRefGoogle Scholar
16.Merkin, J. H. and Needham, D. J., Propagating reaction-diffusion wabes in a simple isothermal quadratic chemical system, J. Engrg. Math. 23 (1989), 343–356.CrossRefGoogle Scholar
17.Merkin, J. H. and Needham, D. J., The development of travelling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic decay and with linear decay, Proc. Roy. Soc. London Ser A 430 (1990), 315–345.Google Scholar
18.Merkin, J. H. and Needham, D. J., The development of travelling waves in a simple isothermal chemical system IV. Quadratic autocatalysis with quadratic decay, Proc. Roy. Soc. London Ser A 434 (1991), 531–554.Google Scholar
19.Merkin, J. H.Needham, D. J. and Scott, S. K., The development of travelling waves in a simple isothermal chemical system I. Quadratic autocatalysis with linear decay, Proc. Roy. Soc. London Ser A 424 (1989), 187–209.Google Scholar
20.Needham, D. J., On the global existence of solutions to a singular parabolic equation arising from the study of autocatalytic chemical kinetics, Z. Angew. Math. Phys. 43 (1992), 471–480.CrossRefGoogle Scholar
21.Needham, D. J. and Merkin, J. H., The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Proc. Roy. Soc. London Ser A 337 (1991), 261–274.Google Scholar
22.Peletier, L. A. and Serrin, J., Uniqueness of positive solutions of semilinear equations in ℝN, Arch. Rational Mech. Anal. 81 (1983), 181–197.CrossRefGoogle Scholar
23.Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Prentice-Hall, Engelwood Cliffs, New Jersey, 1967).Google Scholar
24.Weissler, F. B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29–40.CrossRefGoogle Scholar