Given an equation $P(D)u=0$, where $P$ is a polynomial (not equal to a constant). Here are some basic information about the distributional solution $u$:
- If $P$ has at least one real root, then there exist a (non zero) solution $u \in S'$(temperate distribution);
- It never has a (non zero) solution in $\epsilon'$(distribution with compact support).
These two properties can be checked easily by Fourier transform.
My question is why it always has a distributional solution(non zero) and a $C^{\infty}$ solution(non zero)? For the distributional solution,it's only needed to check that $P(D)C_{0}^{\infty }(\mathbb{R}^{n})$ is a strictly smaller than $C_{0}^{\infty}(\mathbb{R}^{n})$,than Hahn-Banach theorem can be used. For the $C^{\infty}$ solution, I don't know how to deal with it.