# Proving convexity for linear parabolic PDE

1. Jan 14, 2014

### emfert

I have a parabolic PDE of the form $$a\frac{\partial^2 f}{\partial x^2} - b\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} = 0$$, where $$(x,t) \in (-\infty, \infty) \times [0, T]$$. In addition, $\lim_{x \to \infty} f(x,t) = 0$, $\lim_{x\to -\infty} = k$ (a known positive constant), and $f(x,T) = \psi(x)$ where $\frac{d\psi}{dx}<0$, $\frac{d^2\psi}{dx^2}>0$, and $\frac{d^3\psi}{dx^3}<0$. Further, $b \geq a > 0$.

I'm pretty convinced that $\frac{\partial^2 f}{\partial x^2} > 0$ and $\frac{\partial^3f}{\partial x^3} < 0$ but I don't know how to prove it. I'd appreciate any ideas.