First a comment: in the context of **nonlinear wave and Klein-Gordon** equations, the venerable "ABC method" of Cathleen Morawetz is literally "mucking around until you see something". (The A, B, and C refer to the three free function coefficients you can choose for your argument and judicious choices give you good signs for the things you want.)

But most of the time nowadays you do this by only mucking around **after** you see a reason why something you want is likely to be true.

For your example of the **Virial** identity. First: what you wrote down is what I consider an intermediate step of the Virial argument, which would state that a *Virial potential* is convex/concave.

**Linear Schrodinger**

First, think about the case for the linear Schrodinger equation. This models quantum particles without external forces. In the classical limit you have the particles travel on straight lines. In the QM picture the wave packets have centers travelling on straight lines but dispersing slightly. From a central point of view you see that the particles are expected to move in such a way that its radial distance from a fixed point is convex. (Basically centrifugal force.)

Based on this you expect that given a function $a(x) = a(|x|)$ that increases radially, you should see something like
$$ V_a(t) = \int_{\mathbb{R}^N} a(x) |u(t,x)|^2 ~\mathrm{d}x $$
to be convex (in time), at least for suitably chosen weights $a$ that is compatible with the amount of centrifugal force you have.

This $V_a(t)$ is the *Virial potential*. Its first time derivative is equal to the *Morawetz action* (this step holds both for the linear and nonlinear Schrodinger equations)
$$ M_a(t) = \int \nabla a \cdot (\Im u^* \nabla u) ~\mathrm{d}x $$
which, for $a(x) = \frac12 |x|^2$, is what you listed in your original question. But you should keep in mind that the expectation of the Morawetz action having a favorable time derivative is based on the expectation that the Virial potential is convex.

Notice that the convexity of the Virial potential rules out the possibility of mass concentration for the linear Schrodinger equation.

**NLS**

Now supposing your are looking at the nonlinear Schrodinger equation. And you understand that

- ruling out mass concentration can go a long way toward showing that the solution doesn't become singular
- ensuring mass concentration will be a way of showing that the solution does become singular

So you decide to check for your nonlinearity, whether you can still guarantee that the Virial potential is still convex in time (preventing mass concentration) or may it be that for strong enough nonlinearities the potential may turn concave in time (roughly: the presence of an attractive central force that over comes the centrifugal force), which may lead to mass concentration. And then you start computing...

For more similar discussions, I think Tao's *Nonlinear Dispersive Equations* actually gives a lot of "morals" and "motivations" of the sorts of identities and inequalities used.