Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator $P = \partial_t - e^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x} \left( G(x) \cdot\right) $ 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 apprieciated.

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.