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Proving convexity for linear parabolic PDE

  1. Jan 14, 2014 #1
    I have a parabolic PDE of the form [tex]a\frac{\partial^2 f}{\partial x^2} - b\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} = 0[/tex], where [tex](x,t) \in (-\infty, \infty) \times [0, T][/tex]. In addition, [itex]\lim_{x \to \infty} f(x,t) = 0[/itex], [itex]\lim_{x\to -\infty} = k[/itex] (a known positive constant), and [itex]f(x,T) = \psi(x)[/itex] where [itex]\frac{d\psi}{dx}<0[/itex], [itex]\frac{d^2\psi}{dx^2}>0[/itex], and [itex]\frac{d^3\psi}{dx^3}<0[/itex]. Further, [itex]b \geq a > 0[/itex].

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