Can the methods of algebra characterize nonlinear PDE blow-ups? Consider 2 simple differential equations: $x'(t)=x(t)^2, x(0)=1$  and $x'(t)=-x(t)^2, x(0)=1$.
As $t$ goes from $0$ to $\infty$, the first equation ($x(t)=1/(1-t)$) will lead to a finite-time blow-up, while the second ($x(t)=1/(1+t)$) doesn't. Even though, from looking at the algebra, there is only one difference between a plus and a minus. (Luckily, these admit exact solutions. Real-world problems don't.)
Is it therefore correct to think that the methods of algebra (loosely defined, as opposed to traditional analysis of PDEs) will never be able to help us with this kind of question? So far, the algebraic characterizations of PDEs I have seen include some kind of jet bundles (Gromov's partial differential relations, Vinogradov's theory of diffiety which generalizes algebraic geometry etc.). Vinogradov even managed to define spectral sequences and de Rham cohomology, and his school has used them to compute conservation laws / integrable structures of PDEs. Gromov's geometric h-principle is related to developments in fluid dynamics such as Onsager's conjecture. But that is the extent of my knowledge. All the blow-up proofs for PDEs I have seen seem to be mainly analytical in nature.
If you are familiar with such algebraic theories of nonlinear PDEs, do you know a lead that can help characterize PDE blow-ups?
 A: There are classes of PDE's where blow-ups of solutions can be characterised. The examples I am aware of are those of integrable PDE's. To connect to classical algebraic theory, let us focus on so-called finite gap solutions. A typical example of integrable PDE's is given by the $\sinh$-Gordon equations
$$\Delta u=\mp \sinh 2u,$$
where $u$ is a real-valued function defined on the complex plane. Doubly periodic finite gap solutions are given by a linear flow: there exists a (real) linear map from the torus determined by the period lattice into the Jacobian  of the so-called spectral curve. This linear map has to take values inside a real (or quaternionic) component  of the Jacobian in order for $u$ to be real-valued. The solution becomes singular (it has a blow up) where the linear map intersects the $\theta$-divisor inside the Jacobian. For some classes (e.g., the above sinh-Gordon equation with the - sign)  one can prove that you will never intersects the $\theta$-divisor (see Hitchin, Harmonic 2-tori in the 3-sphere, Journal Diff Geo, 1990, Proposition 7.15), while for other classes (e.g. a certain $\cosh$-Gordon) equation you necessarily intersectthe $\theta$-divisor (see Babich-Bobenko, Willmore tori with umbilic lines, Duke Journal,  1993).
