For an $n$th order ODE we always need $n$ boundary conditions (right?). But, as I've seen somewhere, for 2nd order PDEs there are many possible situations and a general answer to the question of how/what boundary conditions are needed doesn't exist. For example, Dirichlet or Neumann boundary conditions give unique solution for elliptic equations, while they don't for hyperbolic equations.
Isn't it really possible for a PDE with some specified boundary conditions to tell easily (with a rule of thump, like for ODEs) if it has a unique solution?
With PDEs is not as simple as with ODEs.
In ODEs, if the equation is of $n$th order, then you need $n$ initial conditions, to guarantee uniqueness.
In the case of hyperbolic PDEs of $n$th order, $$ L_1\cdots L_n u=f, $$ where $L_j$ are first order linear operators, then you need $n$ initial condition, as in the case of the wave equation.
But in the case of elliptic equations, i.e., $L=\Delta^n$, then you need $n$ boundary conditions for $Lu=f$, with $L$ being an operator of order $2n$.