Nonexistence of monotonic solutions in a model of dendritic growth

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 $y\left ( \infty \right ) = \sqrt 2$.