  Despite the strong focus of regularization on ill-posed problems, the general
construction of such methods has not been fully explored. Moreover, many
previous studies cannot be clearly adapted to handle more complex scenarios,
albeit the greatly increasing concerns on the improvement of wider classes. In
this note, we rigorously study a general theory for filter regularized
operators in a Hilbert space for nonlinear evolution equations which have
occurred naturally in different areas of science. The starting point lies in
problems that are in principle ill-posed with respect to the initial/final
data\textendash these basically include the Cauchy problem for nonlinear
elliptic equations and the backward-in-time nonlinear parabolic equations. We
derive general filters that can be used to stabilize those problems.
Essentially, we establish the corresponding well-posed problem whose solution
converges to the solution of the ill-posed problem. The approximation can be
confirmed by the error estimates in the Hilbert space. This work improves very
much many papers in the same line of field.
