What you've found is basically "survivor bias", so it helps for me to describe a bit where the null conditions came about.
Assertion 1: Quasilinear partial differential equations, in general, are too hard to study.
Basically the problem is that the behavior of solutions to partial differential equations depend greatly on the properties of the principal part of the equation. For quasilinear equations the principal part has coefficients that depend on the solution itself. This means that, for example, to understand all solutions of a quasilinear wave equation, you need to understand all wave-type principal symbols. While this is easily done if you are studying locally in time, if you want to get any long-time results this is far out of reach. (I would contend that we don't yet have a complete understanding of the long time behavior of linear wave equations if you allow essentially arbitrary smooth coefficients.)
(There is one exception to this rule for wave-type equations, which is when you have one space and one time dimensions, where the method of characteristics actually do give you a very good understanding of solutions.)
Survivorship: thus works on nonlinear wave equations typically focus on situations where we have either a priori control on the principal symbols, or where we may expect that the principal symbols can be a fortiori controlled (via a bootstrap arguments, say). In the case of "stability" problems, the expectation is that given a specific (smooth) solution to the quasilinear wave equation, that small initial data perturbations will lead to, at least for short time, a solution that is close to the original solution, and hence has approximately the same principal symbol.
In short, because the full problem is too hard, people gravitate toward "small data" problems.
Assertion 2: The small data problem for quasilinear waves in spatial dimension $d \geq 4$ is "easy". (This can also be extended to non-flat backgrounds to a certain extent.)
Via the small data assumption, we can expect that the principal symbol involved will be approximately the linear wave operator on $\mathbb{R}\times\mathbb{R}^d$. Standard linear methods tells us that for the initial value problem, given data in $C^\infty_c(\mathbb{R}^d)$ will lead to solutions that decay in time like $(1+|t|)^{-(d-1)/2}$ or better. When $d \geq 4$, this means that if we approximate
$$\tag{1}\label{eq:linapprox} \Box_{\mathrm{quasilinear}} \phi = \Box \phi + \underbrace{(\Box - \Box_{\mathrm{quasilinear}})\phi}_{\mathrm{Error}(\phi)\cdot \partial^2\phi} $$
The coefficients in $\mathrm{Error}$ is integrable in time, and if initial data is of size $\epsilon$ the product $\mathrm{Error}(\phi)\cdot \partial^2\phi$ is size $\epsilon^2$. So by Duhamel's formula and a Picard-type argument we can at least formally iterate the approximation and expect something to converge.
Survivorship: Since the small data problem is essentially completely solved in $d \geq 4$, people's efforts become focused on the remaining cases.
Assertion 3: The null conditions are/were the next "approachable" cases.
The observation is that the argument in $d \geq 4$ can be largely reproduced as long as we have a way of ensuring that the error terms $\mathrm{Error}(\phi)$ can be guaranteed to decay faster than the expected $(1+|t|)^{-(d-1)/2}$ rate. So between the 1980s and 2000s a lot of people try to study problems using this idea. Note that for this idea to be usable in general, the "improvement in decay" should come from a combination of (a) standard facts about decays of linear waves and (b) structural properties of the nonlinear equations.
For scalar waves, part (b) is essentially solved (there is a simple description of nonlinearities that provide this improvement, and that is the original "null conditions").
Survivorship: the study of part (a) leads to works on non-quasi-diagonal systems of linear wave equations, and to works studying the decay of linear waves on curved backgrounds. Part (b) has also been extended to the case of systems where additional degrees of freedom are present which allow other sorts of non-resonant cancellations. I would say that we understand a lot of what goes into null conditions, but probably there are still plenty to say in special cases.
Additionally, it turns out that the null condition can also be interpreted microlocally as a non-resonant condition. So beyond the original motivation of stability of small data solutions to quasilinear waves, it has also been applied (especially in the semilinear setting) to improved regularity results. (Briefly: in the theory of distributions there is a result that says that one can multiply two distributions if their wavefront sets are disjoint; the wavefront set being a microlocal description of "where the distribution fails to be smooth". A bilinear product obeying the null condition happens to have the property that (roughly speaking), given two solutions of the linear wave equation, the product will only involve pairs with disjoint wavefront sets).
Assertion 4: The studies of the "weak null" conditions came next.
The weak null conditions are rather harder to describe as a mathematical definition, compared to the null condition. The intention, however, is clear: it is a way to capture additional "structural" cancellations that occur in the nonlinearities beyond what was describable as a null condition. To describe it, let us consider the standard Picard iteration. Let $U$ be the solution operator that, given a function $F$ on $\mathbb{R}\times \mathbb{R}^d$, solves
$$ \Box \phi = F $$
with initial data $\phi_0, \phi_1$. Then the iteration for solving the nonlinear problem \eqref{eq:linapprox} sets $\phi^{(0)} \equiv 0$, and generates a sequence $\phi^{(j)}$ with $\phi^{(j+1)} = U( \mathrm{Error}(\phi^{(j)}) \partial^2 \phi^{(j)})$. The null condition provides (note, I am making a lot of simplification here, so am lying quite a bit) sufficient conditions for $U$ to be a contraction mapping, when $(\phi_0, \phi_1)$ are sufficiently small.
The weak null conditions may be described as conditions that ensure the iterated map $\phi^{(j)} \mapsto \phi^{(j+k)}$ is a contraction mapping for some $k > 1$. This says that the nonlinear interactions may have a "large part" provided that it is "nil-potent": that it disappears after finitely many iterations.
Now about equations without null condition.
First, note that it was recognized early on, in the context of one dimensional conservation laws, that null condition is the opposite of the genuine nonlinear condition of Lax, and that in the one dimensional case the system being genuinely nonlinear implies that that no compactly supported initial data can generate global solutions (Fritz John sometime in the early 80s). The proof goes something like this:
- Suppose, for contradiction, that a solution exists for all time, then since the initial data is compactly supported, the solution will eventually (using a similar argument to Riemann invariants) decouple into several families of simple waves.
- Simple waves cannot exist for all time due to shock formation.
Note that the proof doesn't say that "shocks will always form", it says that if nothing else goes wrong, then shocks will eventually take care of business.
I would say that the reason most recent works on wave equations without some sort of null condition or weak null condition focus on the effects of shock formation is because of several reasons:
- Firstly, it is the only mechanism that we have any handle on; in fact, from the physical observations it seems to be the dominant effect generating singularities in nonlinear wave phenomena. (You can "see" shock waves.)
- We still don't fully understand multidimensional shock formation, and our understanding of the multidimensional shock development problem is very rudimentary. (In one dimension, there is a well-established theory in systems of conservation law that provides conditions that ensure unique global weak solutions.) This seems to be a problem that is just barely within reach now.
- Most people studying wave equations (hyperbolic systems) are motivated somewhat from applications. It was a great boom 30 years ago when it was discovered that a lot of the natural equations that come up in physics exhibit the null condition. Currently the emphasis on shocks is driven largely by our desire to understand fluid dynamics.
Returning to your original questions:
There are certainly works on quasilinear waves that do not obey the classical null conditions. (My paper with Abbrescia being one such.) But you are asking the wrong question, by lumping things into "null condition versus no-null condition". The better way to classify results for the global Cauchy problem in quasilinear wave theory is to split them into (A) results that ensure global-in-time existence (B) results that tries to ensure singularity formation and (C) results that tries to classify the singularity types.
(A) certainly still has life left beyond the null conditions, while (B) and (C), as you observed, are currently dominated by results about shock formation; these are in part due to the reasons I observed in the previous section.
In addition, there are also problems concerning the local Cauchy problem (minimum regularity for local wellposedness, minimum regularity to control time of existence of smooth solutions); for reasons I discussed above, null condition does play a role here in improving regularity, but the problem without structure assumptions are equally studied. (Lindblad, Smith-Tataru, Q. Wang, etc.)
Off hand I cannot think of any conjectures that meet your requirement. (There is the big open problem of shock development, but I don't think it is formulated as a "conjecture" per se.) But that is probably because conjectures are not formed in vacuum, but formed through experience. And most interesting physical wave equations that folks work on just naturally have null condition, and so demanding to only look at conjectures concerning wave equations without null conditions is really limiting.
