Do physicists do 'real' PDEs?And I mean real, well-defined, well-posed and not out of this world?

not sure exactly what u have in mind?

The Schrodinger equation is an interesting PDE. It is valid approximation when v<<c. It is directly linked/derived from standard momentum (at least in some of its forms). Standard momentum for a particle with mass is zero when a particle is at rest. This is also where the modern physics standard matter wave is infinite. The de Broglie matter wave is derived from standard momentum.

"De Broglie had an extremely strong and concrete physical justification for the infinite wavelength of matter waves, corresponding to the body at rest.”—H. Chauhan et al.

So the Schrodinger equation can only be a good approximation when v<<c and v>0. So it dose not really hold for rest-mass particles, nor for fast moving particles. Modern physics have partly tricked themselves around the zero momentum problem, by introducing two momentum formulas that they not have any direct connection between. That is an additional and separate formula for the momentum of photons, that series of researchers without discussing it even uses for particles with rest-mass, (at the same time they assume photons do not have mass).

These issues are avoided when one release that the de Brolige wave is just a mathematical derivative of the true matter "wave", namely the Compton wave. Also the standard momentum is a derivative of what we can call the Compton momentum. For Compton momentum one have well defined rest-mass momentum, in addition to kinetic and total momentum. The photon momentum in standard physics we get from the Compton momentum when v=0. Here we get one formula for momentum that holds for all particles, also photon-photon collisions. At photon-photon collision the photons stand still for a Planck second and are mass, and have momentum.

Another issue is if space is quantized on not, in my view space is not quantized, it is continuous divisible void. If it is quantized can we then use PDE's? "PDEs are equations that involve rates of change with respect to continuous variables. (wiki) " ? Is this where

QPDEs comes in?