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{equation}
\begin{cases}
\phi_{tt}+\Delta\phi +|\phi|^{p-1}\phi=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}
   \end{equation}
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.
 A: I'm just copy-pasting a paragraph due to Terence Tao on this matter:
If the data class is subcritical with respect to scaling, one can use the scaling symmetry to trade between time of existence and size of initial data; thus if one establishes a local wellposedness at a fixed time (say up to time $T = 1$) for data with small norm, then one can often also establish local wellposedness at a small time for large data; typically the time of existence will be proportional to some negative power of the norm of the data. Conversely, if the data class is supercritical with respect to scaling (or more generally is lower than the invariant norm associated to another symmetry), then it is likely that there is a significant obstruction to obtaining a wellposedness theory below that regularity, and one also expects the wellposedness theory at that regularity to be rather delicate. The reason for this is that if the regularity is below the invariant regularity, then one can use the symmetry to convert bad behaviour arising from large initial data at some time $t > 0$ to bad behaviour arising from small initial data at some time less than or equal to $t$, where "large" and "small" are measured with respect to the regularity $H^s(R^d)$. Since large initial data would be expected to display bad behaviour very quickly, one then expects to give examples of arbitrary small initial data which displays bad behaviour arbitrarily quickly.
We refer to regularities $s > s_c$, above the critical norm as subcritical, and regularities $s < s_c$, below the critical norm as supercritical. The reason for this inversion of notation is that higher regularity data has
better behaviour, and thus we expect subcritical solutions to have less pathological behaviour than critical solutions, which in turn should be better behaved than supercritical solutions. The other scalings also have their own associated regularities;
the Galilean symmetry and pseudoconformal symmetry preserve the $L^2 (\mathbb{R}^n)$ norm, whereas the Lorentz symmetry and conformal symmetries are heuristically
associated to the $\dot H^{1/2}(\mathbb{R}^d) \times \dot H^{-1/2}(\mathbb{R}^d)$ norm.
Reference: T. Tao, Nonlinear Dispersive Equations, local and global analysis, CBMS, Regional Conference Series in Mathematics, number 106.
