# What does it really mean for a wave equation to be critical?

I am trying to understand intuitively the concept of criticality in general for Wave equations. For example, consider the cauchy problem of semi-linear equation $$$$\begin{cases} \phi_{tt}+\Delta\phi +|\phi|^{p-1}=0; \quad x\in \mathbb{R}^n\\ \phi(x,0)=f(x)\in H^1(\mathbb{R}^n), \quad \phi_t(x,0)=g(x) \in L^2(\mathbb{R}^n) \end{cases}$$$$

I am trying to understanding what does it mean for the exponent $$p$$ defined in the equation above to be critical exponent, thereby making it critical.

I understand that $$p$$ must satisfy certain condition for it to be critical (subcritical or even supercritical) at the level of certain Sobolev space $$H^s$$ say, if the Cauchy problem above stays invariant under some scalling transformation. And that does not mean criticality for blow-up, but rather for minimal smoothness condition for the existence (local) of solution.

My questions:

(a) Why do we need to study criticality of certain wave equation?

(b) What are the differences between the criticality for blow-up and that of the equation.

(c) What condition must $$p$$ satisfy to be considered as a critical exponent at least for blow-up.

Any hint that clear the way to understand the difference is highly welcome.