Nonexistence of monotonic solutions in a model of dendritic growth
Author:
William C. Troy
Journal:
Quart. Appl. Math. 48 (1990), 209-215
MSC:
Primary 80A20; Secondary 34C11
DOI:
https://doi.org/10.1090/qam/1052131
MathSciNet review:
MR1052131
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Abstract: A simple model for dendritic growth is given by ${\delta ^2}\theta''' + \theta ’ = \cos \left ( \theta \right )$. For $\delta \approx 1$ we prove that there is no bounded, monotonic solution which satisfies $\theta \left ( { - \infty } \right ) = - \pi /2$ and $\theta \left ( \infty \right ) = \pi /2$. We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely $y'''+ y’ = 1 - {y^2}/2$. We prove that there is no monotonic solution which satisfies $y\left ( { - \infty } \right ) = - \sqrt 2$ and $y\left ( \infty \right ) = \sqrt 2$.
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R. C. Brower, D. A. Kessler, J. Koplik, and H. Levine, Geometrical approach to moving-interface dynamics, Phys. Rev. Letters 51, 1111–1114 (1983)
R. C. Brower. D. A. Kessler, J. Koplik, and H. Levine, Geometric models of interface evolution, Phys. Rev. A 29 (3), 1335–1342 (1984)
R. F. Dashen, D. A. Kessler, H. Levine, and R. Savit, The geometrical model of dendritic growth; The small velocity limit, Physica D 21, 371–380 (1984).
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theo. Phys. 55, 356–369 (1976)
M. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, preprint
D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Physica D 19, 89–111 (1986)
G. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, I. Derivation of basic equations, Acta Astronautica 4, 1117–1206 (1977)
W. C. Troy, The existence of steady solutions of the Kuramoto-Sivashinsky equation, J.D.E. 82 (2), 269–313 (1989)
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© Copyright 1990
American Mathematical Society