I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it's fine if it's a partial differential equation.
At first I thought it would be easy to either come up with an example or else find one by searching google/wiki/arxiv; but now I am not so sure.
I have a thing for non-smooth functions, and it just bothers me that I don't even know a single example of this type of differential equation. Thanks!
Best Answer
There are already first order linear partial differential equations with smooth coefficients which do not admit any smooth solutions.
Hans Lewy produced the first example of such a PDE. The equation reads $$\left[-i\partial_x+\partial_y-2(x+iy)\partial_z\right]u(x,y,z)=f(x,y,z),\qquad(x,y,z)\in\mathbb R^{3}.$$ The equation does not have distribution solutions in any neighbourhood of any point in $\mathbb R^3$ provided $f=f(x,y,z)$ is not a real analytic function (it can be smooth though).
The original paper by Lewy is nice, clear and less than 4 pages long (freely available here).