  Regularization methods have been recently developed to construct stable
approximate solutions to classical partial differential equations considered as
final value problems. In this paper, we investigate the backward parabolic
problem 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 general
ill-posed. The ill-posedness is caused by catastrophic growth in the
representation of solution, then independence of solution on data makes
computational procedures impossible. Therefore, we contribute to this
interesting field the study of the stable approximation solution via the
modified quasi-boundary value method based on nonlinear spectral theory, which
slightly extends the results in many works. Our main focus is to qualitatively
prove the stability and convergence rate where semi-group technicalities are
highly useful.
