Growth condition in differential equation and vanishing solution at boundaries

Question

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary conditions at z_i \element (0,1), because the resulting non-constant coefficients of the transformed diffuion-equation vanish sufficiently fast at the boundaries." I would love to see an example of a (P)DE with a 'standard growth condition' (what is it?) and find out if "[...] coefficients [...] vanish [...] at the boundaries" means that one could simply require the solution to be zero at the boundaries.