A singular perturbation problem in needle crystals (Q909861)

The equation \[ \epsilon \theta '''(x)+\theta '(x)=\cos \theta (x),\quad 0\leq x<\infty;\quad \epsilon >0; \] arises in the problem of dentritic solidification. The authors prove that it has a unique monotonic solution satisfying \[ \theta (0)=0;\quad \lim_{x\to \infty}\theta (x)=\pi. \] The existence of needle-crystal solutions depends on whether \(\theta ''=0\). The authors show that this is not the case and give an asymptotic estimate of \(\theta ''(0)\) when \(\epsilon\) tends to zero.