As it is clearly explained here by L. N. Trefethen, not every linear partial differential equation has a solution (not even in the sense of distribution theory). The referred example is the one given by Hans Lewy in 1957 in Annals of Mathematics.
There exists a $C^\infty$ real function $f=f(t)$ such that the nonhomogeneous linear equation
$ \displaystyle \frac{\partial u}{\partial x} + i \frac{\partial u}{\partial y} -2i(x+iy) \frac{\partial u}{\partial t} = f(t),$
has no solution.
As stated by K. O. Friedrichs:
Once it is said, it is clear to everybody; but Lewy was the first to say it!
Other than the PDEs in the first link or in the references therein, are there other known linear partial differential that fail to have a solution?