The effects of transport on the maximum principle

We study the qualitative effects of the transport terms on the maximum principle for a family of one-dimensional linear second order boundary value problems. We mainly focus on sign changing transport functions, denoted by B(x). There is a huge difference between the cases when B is non-increasing or non-decreasing. It turns out that when B is non-increasing the maximum principle holds when the amplitude of B is sufficiently large, whereas it fails to be true if B is non-decreasing.