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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}=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.

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