The PDE $\partial_t = \kappa\partial_x\partial_y f$, where $\kappa$ is a constant, came up in something I was working on. Boundary conditions are vanishing at infinity. At first, it looks heat equation-ish, so naturally one would try separating variables and Fourier transforming in a similar way to the derivation of the heat kernel. However, doing this gives for a fundamental solution $$ f(x,y, t) = \frac{1}{2\pi}\iint_{-\infty}^\infty \exp(-\sigma tk_xk_y +ik_x x+i k_y y)dk_x dk_y, $$ an integral that doesn't seem to converge. It might using a more generalized sense of the Fourier transform, but I'm not well-versed in Fourier transforms over distributions.
Some other notable aspects of the equation:
The differential operator is linear and shift invariant. It is also scale invariant for isotropic scalings. Additionally, the operator conserves the integral of $f$. This means we can if necessary assume $f$ has unit norm, zero mean, and can isotropically scale its variance however we want.
The differential operator is not isotropic. It can be diagonalized by the change of variables $u = x + y$, $v = x - y$. This gives $\partial_t f = \sigma(\partial_u^2 - \partial_v^2)f$. In particular, the equation tends to narrow the initial condition along $x = y$ and spread it along $x = -y$ going forward in time (and the reverse going backwards).
All solutions with finite initial variance have finite-time blowup, both forwards and backwards in time. In particular, if $\sigma^2(\theta,t) = \iint (x\cos\theta+y\sin\theta)^2 f(x,y,t)dxdy$ denotes the variance in the direction $\theta$, then $\sigma^2(\theta,t) = \sigma^2(\theta,0) - 2\kappa t\sin(2\theta)$. Thus there is always a direction in which the variance vanishes in finite time both forwards and backwards.
The bivariate normal distribution $$ n(x,y,t) = \frac{1}{2\pi\sqrt{1-t^2}}\exp\left[-\frac{x^2 + y^2 - 2xyt}{2(1-t^2)}\right] = \left[\frac{e^{-\frac{(x-y)^2}{4(1-t)}}}{\sqrt{2\pi(1-t)}}\right]\left[\frac{e^{-\frac{(x+y)^2}{4(1+t)}}}{\sqrt{2\pi(1+t)}}\right] $$ is a solution for $\kappa = 1$. It evolves to $\delta(x-y)\exp(-x^2/2)/\sqrt{2\pi}$ at time $t = 1$ and $\delta(x+y)\exp(-x^2/2)/\sqrt{2\pi}$ at $t = -1$. This function looks very heat kernel-ish, and might be related to the fundamental solution.
So, any help in finding this fundamental solution? Is there a method of finding it I don't know about? Does the finite-time blowup mean I'm barking up the wrong tree?