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We consider the identification of nonlinear diffusion coefficients of theform $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations.Uniqueness for this inverse problem is established under very generalassumptions using partial knowledge of the Dirichlet-to-Neumann map. The proofof our main result relies on the construction of a series of appropriateDirichlet data and test functions with a particular singular behavior at theboundary. This allows us to localize the analysis and to separate the principalpart of the equation from the remaining terms. We therefore do not requirespecific knowledge of lower order terms or initial data which allows to applyour results to a variety of applications. This is illustrated by discussingsome typical examples in detail.

On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms