TITLE: When can one expect a classical solution of a PDE?
QUESTION [16 upvotes]: When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer to?
Consider for example, in Folland's Introduction to Partial Differential Equations, Chapter 3, which is devoted to the solution of the Dirichlet and Neumann problems for the Laplacian by the method of layer potentials. 

OK, one may say that it quite depends on the context. It may be too vague to ask such questions here. Let me "rephrase" the question a little bit:

When can one expect a classical solution of a PDE? 
Is there a rule of thumb for when a PDE may fail to have a classical solution, such that one has to look for a weak one?

I think it may be convenient to restrict the attention to the three classical types of PDEs; namely, the Poisson equations, heat equations and wave equations.

REPLY [3 votes]: Instead of considering the three types of PDE in the OP, one can restrict the attention to the constant-coefficient operators. Then one has the following nice result, which I learned from Folland's Introduction to Partial Differential Equations (Chapter 1 (1.54)):

If $L$ is a differential operator with constant coefficients on ${\mathbb R}^n$ and $f\in C^{\infty}_c({\mathbb R}^n)$, there exists $u\in C^{\infty}({\mathbb R}^n)$ such that $Lu=f$.

This is an immediate corollary of the existence of a fundamental solution:
The Malgrange-Ehrenpreis Theorem
Every differential operator $L$ with constant coefficients has a fundamental solution.

Note:
With a fundamental solution $K$ in hand, we can solve the equation $Lu=f$ not only when $f\in C^{\infty}_c$ but when $f$ is any distribution with compact support; the solution $u$ will then be a distribution.