Question

To elaborate on Steve Huntsman's comment, I remember reading the following on Terence Tao's blog: there exist PDE that can simulate Newtonian mechanics, and using such a PDE and the correct initial conditions it is possible, in principle, to simulate an arbitrary analog Turing machine. So a general-purpose algorithm to determine even the qualitative behavior of an arbitrary PDE cannot exist because such an algorithm could be used to solve the halting problem.