  We consider the identification of nonlinear diffusion coefficients of the
form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations.
Uniqueness for this inverse problem is established under very general
assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof
of our main result relies on the construction of a series of appropriate
Dirichlet data and test functions with a particular singular behavior at the
boundary. This allows us to localize the analysis and to separate the principal
part of the equation from the remaining terms. We therefore do not require
specific knowledge of lower order terms or initial data which allows to apply
our results to a variety of applications. This is illustrated by discussing
some typical examples in detail.
