The stability of vector renewal equations pertaining to heterogeneous chemical reaction systems

Authors:
A. E. DeGance and L. E. Johns

Journal:
Quart. Appl. Math. **34** (1976), 69-83

MSC:
Primary 82.45

DOI:
https://doi.org/10.1090/qam/449376

MathSciNet review:
449376

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a class of vector renewal equations arising in the theory of spatially nonuniform chemical reaction processes, stability conditions are given in terms of the physicochemical operators **D** and **K**. In particular, we provide conditions for the stability of an initially uniform, multicomponent film bounding a planar catalytic surface, and for the asymptotic stability of a reaction system composed of a population of catalyst particles. We obtain such results by identifying a sup norm which is naturally induced from the factors that symmetrize **D** and/or **K**. Further, we exhibit conditions under which a special class of vector renewal equations has positive solutions.

**[1]**Richard Bellman,*Introduction to matrix analysis*, Second edition, McGraw-Hill Book Co., New York-Düsseldorf-London, 1970 (Russian). MR**0258847****[2]**Richard Bellman and Kenneth L. Cooke,*Differential-difference equations*, Academic Press, New York-London, 1963. MR**0147745****[3]**A. E. DeGance,*Diffusion--heterogeneous chemical reaction interference in multicomponent systems*, Ph.D. dissertation, University of Florida, 1974**[4]**A. E. DeGance and L. E. Johns,*The sensitivity of the selectivity to transport parameters in batch heterogeneous chemical reaction systems*, Chem. Eng. J.**7**, 227-244 (1974)**[5]**Rodney D. Driver,*Some harmless delays*, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 103–119. MR**0385277****[6]**R. J. Duffin,*A minimax theory for overdamped networks*, J. Rational Mech. Anal.**4**(1955), 221–233. MR**0069030****[7]***Handbook of mathematical functions*, N. B. S. AMS 55, Government Printing Office, Waashington, D. C., 1965**[8]**J. L. Hudson,*Transient multicomponent diffusion with heterogeneous reaction*, A.I.Ch.E. J.**13**, 961-964 (1967)**[9]**L. E. Johns and X. B. Reed,*Diffusion--heterogeneous chemical reaction in multicomponent systems*.*I*.*The penetration theory for single catalyst particles in a dilute turbulent suspension*, Chem. Engng. Sci.**28**, 275-288 (1973)**[10]**L. E. Johns and X. B. Reed,*On imperfect contacting in batch heterogeneous chemical reaction systems*, Chem. Engng. Sci.**28**, 1591-1609 (1973)**[11]**Richard K. Miller,*Nonlinear Volterra integral equations*, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR**0511193****[12]**J. A. Nohel,*Some problems in nonlinear Volterra integral equations*, Bull. Amer. Math. Soc.**68**(1962), 323–329. MR**0145307**, https://doi.org/10.1090/S0002-9904-1962-10790-3**[13]**Georgi E. Shilov,*An introduction to the theory of linear spaces*, Translated from the Russian by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR**0126450**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
82.45

Retrieve articles in all journals with MSC: 82.45

Additional Information

DOI:
https://doi.org/10.1090/qam/449376

Article copyright:
© Copyright 1976
American Mathematical Society