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The present paper analyzes a spectral regularization of a time-reversedreaction-diffusion problem with globally and locally Lipschitz nonlinearities.This type of inverse and ill-posed problems arises in a variety of real-worldapplications concerning heat conduction and tumour source localization. Inaccordance with the weak solvability result for the forward problem, we focuson the inverse problem with high-order Sobolev-Gevrey smoothness and withSobolev measurements. As expected from the well-known results for the linearcase, we prove that this nonlinear spectral regularization possesses alogarithmic rate of convergence in a high-order Sobolev norm. The proof can bedone by the verification of variational source condition; this way validatessuch a fine strategy in the framework of inverse problems for nonlinear partialdifferential equations. Ultimately, we study a semi-discrete version of theregularization method for a class of reaction-diffusion problems withnon-degenerate nonlinearity. The convergence of this iterative scheme is alsoinvestigated.

Convergence of a spectral regularization of a time-reversed reaction-diffusion problem with high-order Sobolev-Gevrey smoothness