This is related to: Partial Differential Equations: Fourier Transform in Space and Time?, and The equation $r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s}.$ Possible connections in physics and math?
Consider the function, or distribution:
$$\Psi(r,x)=\exp {\frac{r}{\log x}}$$
which satisfies the linear second order parabolic partial differential equation:
$$r \frac{\partial ^2\Psi(r,x)}{\partial r^2}=-x \frac{\partial \Psi(r,x)}{\partial x} \tag{1}$$
Let $\hat \Psi$ be the Mellin transform* of $\Psi$ w.r.t. $x.$ Then $\hat \Psi $ satisfies the below linear third order partial differential equation:
$$\hat \Psi(r,s)=2 \sqrt{\frac{r}{s}}K_1(2\sqrt{r s})$$
$$r^2 \frac{\partial ^3\hat \Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \hat \Psi(r,s)}{\partial s} \tag{2}$$
where $K_1$ is the modified Bessel function of the second kind.
This implies that the equations $(1)$ and $(2)$ are connected through the Mellin transform.
I'm aware that the Laplace transform is more often used in cases where the PDE is parabolic. Furthermore, I'm struggling with the physical interpretation of $(r,s)$-space. Would this be $r$ (space) and $s$ (complex time) or something? But it's hard to get a read on the physical interpretation of the Mellin transform. I do know that the Mellin transform is scale invariant...another thought I had was to view $(1)$ in terms of a heat equation with $r$ (space) and $x$ (time).
Is there an interpretation here, even if it doesn't make sense physically? Am I on the right track with my own interpretation (i.e. thinking of $(1)$ as a heat equation etc.)?
*More details on $\hat \Psi$ and $\Psi$ being connected by the Mellin transform:
$$\int_0^1 x^{s-1} \exp \frac{r}{\log x}~dx =\hat \Psi(r,s)$$