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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The qualitative analysis of a dynamical system
of differential equations arising from the
study of multilayer scales on pure metals II


Author: R. L. Baker
Journal: Proc. Amer. Math. Soc. 127 (1999), 753-761
MSC (1991): Primary 34C35, 70K05
MathSciNet review: 1469394
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Abstract: We provide a qualitative analysis of the $n$-dimensional dynamical system:

\begin{displaymath}\dot q_i=-\sum _{j=1}^n \frac{a_{ij}}{q_j^k},\quad q_i(t)>0,\qquad i=1,\dots, n, \end{displaymath}

where $k$ is an arbitrary positive integer. Under mild algebraic conditions on the constant matrix $A=(a_{ij})$, we show that every solution $\mathbf q(t)$, $t\in[0,a)$, extends to a solution on $[0,+\infty)$, such that $\lim _{t\to+\infty} q_i(t)=+\infty$, for $i=1,\dots, n$. Moreover, the difference between any two solutions approaches $0$ as $t\to+\infty$. We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.


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Additional Information

R. L. Baker
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: baker@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04563-3
PII: S 0002-9939(99)04563-3
Keywords: Differential equations, dynamical systems, nonlinear dynamical systems, cooperative dynamical systems
Received by editor(s): June 20, 1994
Received by editor(s) in revised form: June 16, 1997
Communicated by: David Sharp
Article copyright: © Copyright 1999 American Mathematical Society