As a follow up to: Fourier Transform of a PDE in 2 spatial variables.
I wish to classify the right side of the equation
$\partial_t u = \partial_x^2 u + x \partial_y u,$
viewed as an operator. From what I understand, the principal part of the operator is $\partial_x^2 $ with principal symbol $-\xi^2$, and in $\mathbb{R}^2 - \{0\}, \implies -\xi^2 \neq 0$ which suggests that the operator is elliptic, is this the case?