Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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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DOI: https://doi.org/10.1090/qam/1052131
Article copyright: © Copyright 1990 American Mathematical Society


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