I'm assuming that non-mathematical subjects, like physics, don't count --- there the heat, wave, Schrödinger, KdV, water wave equation, Navier-Stokes, Helmholtz, ..., equations are all fairly important objects. In fact most of the PDE I could name would be related to physics in some way. I would say that most PDE are in this direction.
In some sense, the entire field of complex analysis comes down to genuinely understanding solutions to one PDE; complex analysis, I think you'd agree, is a pretty big field, with plenty of applications of its own.
A number of tools have been produced by PDE which are of universal appeal in analysis. For example, the Fourier transform, which has a broad range of applications in analysis, not to mention generalizations, e.g. the Gelfand map, was developed as a tool to solve the wave equation. Another is the convolution (which I'm assuming is also from PDE) and along with it a variety of dense functions, nice partitions of unity, and so on, along with notions of convergence which are also very useful in a variety of contexts. Things like the Poisson kernel and the Hilbert transform have become prototypical examples in integral operators.
PDE in general are rather hard, and so any particular PDE is likely to be rather narrow in scope. So many of the things of greatest interest to come out of it are tools to solve problems rather than necessarily specific solutions.
