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Regularization methods have been recently developed to construct stableapproximate solutions to classical partial differential equations considered asfinal value problems. In this paper, we investigate the backward parabolicproblem with locally Lipschitz source:$\partial_{t}u+\mu\left(t\right)\mathcal{A}u\left(t\right)=f\left(t,u\right)$where$\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset\mathcal{H}\to\mathcal{H}$is a positive, self-adjoint and unbounded linear operator on the Hilbert space$\mathcal{H}$. The problem arises in many applications, but it is in generalill-posed. The ill-posedness is caused by catastrophic growth in therepresentation of solution, then independence of solution on data makescomputational procedures impossible. Therefore, we contribute to thisinteresting field the study of the stable approximation solution via themodified quasi-boundary value method based on nonlinear spectral theory, whichslightly extends the results in many works. Our main focus is to qualitativelyprove the stability and convergence rate where semi-group technicalities arehighly useful.

Stability and error estimates of a general modified quasi-boundary value method via a semi-linear backward parabolic equation