Well posedness of PDEs of the form $u_{tt} + \left[ P ( \partial_x)\right]^2u=0$

Question

Let $P$ be a polynomial with real coefficients. I want to find the necessary&sufficient conditions on $P$ in order for the problem $$u_{tt} + \left[ P ( \partial_x)\right]^2u=0$$

with

• $u (0,x) = f(x)$
• $ u_t(0,x) = 0$

to be well-posed for every positive period of the data $f$.

I am tempted to decompose the PDE to the form $$P_1(\partial_t,\partial_x) \cdot \overline{P_1(\partial_t,\partial_x)}u=0$$ where $P_1(\partial_t,\partial_x) = \partial_t+iP(\partial_x)$ and $\overline{P_1(\partial_t,\partial_x)} =\partial_t - iP(\partial_x) $ is it's complex conjugate. But I don't know how to take it from there...