# Convexity of solutions of parabolic PDE

I consider the parabolic PDE of the form: $$\frac{\partial u}{\partial t} = a \frac{\partial^2 u}{\partial x^2} + b \frac{\partial u}{\partial x} + f(x) u,$$ where $$a, b$$ are constants. Initial condition is $$u(x, 0) = \phi(x)$$, where $$\phi(x)$$ is a convex function.

I wonder about the convexity of $$u(x, t)$$. I want to know what is the dependence between the form of the $$f(x)$$ and the convexity of $$u(x, t)$$. In other words I want to know if convexity is preserved for convex initial condition.

Can anyone recommend me some mathematical techniques that I can use in this type of problem.