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Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator $$ P = \partial_t - \mathrm{e}^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x} \big( G(x) \cdot\big), $$ where $\beta > 0$ and $G$ is smooth with $(x,t) \in \mathbb{R} \times [0, \infty)$.

I'm guessing not, as the equation "behaves" more and more like a hyperbolic equation as $t \to \infty$. Any references would be greatly appreciated.

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Since your equation is uniformly parabolic for $t\in [0,T]$ ($T$ bounded), most of the arguments should carry on... – Thomas Richard Apr 29 '13 at 14:23
What do you mean by the maximum principle? If you want the maximum of the solution to be decreasing in time, then it would not be true (even replacing the factor $e^{-\beta t}$ by $1$). If you want a comparison principle saying that if one solution is initially larger than another, then the order is preserved by time, then that will be true. If you want a bound on the $L^\infty$ norm for a solution $u$, then $||u(\cdot,t)||_{L^\infty} \leq e^{-(\min G_x)t} ||u(\cdot,0)||_{L^\infty}$ even if the second order term wasn't there. – Luis Silvestre Apr 30 '13 at 16:50

Smoothness on $G$ may not be enough to ensure that a maximum principle can be obtained. If $G$ or $G_x$ blows up as $|x|\to\infty$ then I anticipate a problem. I advise looking at Protter and Weinberger (1967), Walter (1970) and Lieberman (1996) for additional references.

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