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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.

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