A generalized filter regularization result for some nonlinear evolution equations in Hilbert spaces

TL;DR

This paper develops a comprehensive theory for filter regularization of nonlinear evolution equations in Hilbert spaces, providing a unified framework for stabilizing and solving ill-posed problems like nonlinear elliptic and parabolic equations.

Contribution

It introduces a general class of filter regularized operators applicable to complex nonlinear evolution equations, extending previous methods and establishing convergence and error estimates.

Findings

Derived general filters for stabilization of ill-posed problems

Established well-posed problems with solutions converging to original solutions

Provided error estimates confirming approximation accuracy

Abstract

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.