I'll start off with the following example for $u(x,t):\mathbb{C}^2 \to \mathbb{C}$: $$\begin{align} u_t&=x^2 u_x, \\ u(x,0)&=f(x), \end{align}$$ where $f$ is an entire function. The general solution of this problem is $$u(x,t)=f \left(\frac{x}{1-t x} \right), $$ and I'm interested in finding out which initial conditions $f$ lead to an entire solution $u$.
- Obviously, if $f \equiv c$ is a constant function, the solution is $u \equiv c$, which is entire as well.
- If $f$ is a non-constant rational function then the formula for the the solution implies the existence of singularities for $u(x,t)$.
- If $f$ is not a rational function it is known that it has an essential singularity at infinity, in which case $u(x,t)$ will also have those on the hyperbola $tx=1$.
All in all, the only entire initial conditions $f(x)$, which lead to entire solutions $u(x,t)$ are the constant ones.
Notice in contrast that the problem $$\begin{align} u_t=x u_x, \\ u(x,0)=f(x), \end{align} $$ has solution $$u(x,t)=f\left( \mathrm{e}^t x \right) $$ which is entire for any entire $f$.
Here is my question: Given a first order linear homogeneous evolution equation for a function $u(\mathbf{x},t):\mathbb{C}^n \times \mathbb{C} \to \mathbb{C}$ of the form $$\begin{align} u_t&=\sum_{k=1}^n a_k(\mathbf{x}) u_{x_k}\\ u(\mathbf{x},0)&=f(\mathbf{x}) \end{align} $$ with entire coefficients $\{a_k\}_{k=1}^n$, is there a way to determine which initial conditions $f$ will lead to entire solutions $u$? I know that all constant functions satisfy this condition, but when are there others? what are they in this case?
Thank you!