Is anything known about the finite speed of propagation of wave-like nonlinear PDE:

$$u_{tt} - \Delta \left(u^p\right) = 0$$

when say $p > 1$?

That is given initial data $u(x,0) = w_1(x)$ and $u_t(x,0) = w_2(x)$ for $w_1,w_2 \in C_c(R^n)$, if the solution is compactly supported as well? Do energy functions work for these PDEs as well like in the case of wave equation $p = 1$?

Any help/reference is appreciated, thank you.