  The present paper analyzes a spectral regularization of a time-reversed
reaction-diffusion problem with globally and locally Lipschitz nonlinearities.
This type of inverse and ill-posed problems arises in a variety of real-world
applications concerning heat conduction and tumour source localization. In
accordance with the weak solvability result for the forward problem, we focus
on the inverse problem with high-order Sobolev-Gevrey smoothness and with
Sobolev measurements. As expected from the well-known results for the linear
case, we prove that this nonlinear spectral regularization possesses a
logarithmic rate of convergence in a high-order Sobolev norm. The proof can be
done by the verification of variational source condition; this way validates
such a fine strategy in the framework of inverse problems for nonlinear partial
differential equations. Ultimately, we study a semi-discrete version of the
regularization method for a class of reaction-diffusion problems with
non-degenerate nonlinearity. The convergence of this iterative scheme is also
investigated.
