Consider the PDE

$$P(x, y) \frac{\partial^{2} u}{\partial x^{2}}+e^{x^{2}} e^{y^{2}} \frac{\partial^{2} u}{\partial x \partial y}+Q(x, y) \frac{\partial^{2} u}{\partial y^{2}}+e^{2 x} \frac{\partial u}{\partial x}+e^{y} \frac{\partial u}{\partial y}=0$$

where $P$ and $Q$ are polynomials in two variables with real coefficients. Then which of the following is true for all choices of $P$ and $Q$ ?

1. There exists $R>0$ such that the $P D E$ is elliptic in $\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}>R\right\}$

2. There exists $R>0$ such that the $P D E$ is hyperbolic in $\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}>R\right\}$

3. There exists $R>0$ such that the $P D E$ is parabolic in $\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}>R\right\}$

4. There exists $R>0$ such that the $P D E$ is hyperbolic in $\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} <R\right\}$