To add to what the Anonymous Mathematician said (so again, this has to do with United States)
If you have not already decided on a specialisation, you should certainly not try to shoehorn yourself into one just for the sake of applying for graduate school. If you have already decided on a specialisation, however, you should certainly specify it and try to talk to faculty members before and/or during the admission process. This is not so much to improve your chances at the school per se; this is mostly to find out whether that school would be a good fit for what you want to study.
When I was applying for graduate schools I had just one such experience. I was already quite sure about what I wanted to study, and so contacted some faculty members at places where I applied to who may be good advisors. One of them told me that (a) I have a pretty strong application and they will probably admit me (b) But he is personally busy with his existing students and won't want to take another one (c) There's no one in the same department that he sees as having similar enough interests to be a good advisor for me and (d) The department is running a bit low on money so unless I get a fellowship somehow I will have a hellish teaching load.
In the end I took an offer from another school.
For emphasis, however, what I mean by specialisation is very narrow: "partial differential equations" is still too broad. For my advice above to apply you need to be able to say which type of PDEs you are interested it (elliptic, parabolic, evolutionary, dispersive, transport, kinetic theory, fluids, and/or optimal control to list a few) and to be able to hold an informed (not necessarily expert) discussion on the subject and why you want to study it.
For other countries the system can vastly differ; this is especially so for degree programs which require a Masters or equivalent degree for admission. Many of those programs require submission of a research proposal and admission is contingent on the research being likely to be able to be conducted at said university and that a suitable advisor can be found for your research proposal. But for these kinds of degrees the opposite of the expectation described by Anonymous Mathematician holds: the incoming students are expected to be sufficiently educated (by the Masters degree) to make informed decisions and are also expected to have a good idea what is involved in pursuing PhD research and know what they want to work on. But you can general tell the expectations by reading up on the qualifications for admission and on what the departments/universities expect on the application form.
