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?