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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II
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by R. L. Baker PDF
Proc. Amer. Math. Soc. 127 (1999), 753-761 Request permission

Abstract:

We provide a qualitative analysis of the $n$-dimensional dynamical system: \[ \dot q_i=-\sum _{j=1}^n \frac {a_{ij}}{q_j^k},\quad q_i(t)>0,\qquad i=1,\dots , n, \] 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.
References
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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
  • Received by editor(s): June 20, 1994
  • Received by editor(s) in revised form: June 16, 1997
  • Communicated by: David Sharp
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 753-761
  • MSC (1991): Primary 34C35, 70K05
  • DOI: https://doi.org/10.1090/S0002-9939-99-04563-3
  • MathSciNet review: 1469394