A nonlinear parabolic equation backward in time: Regularization with new error estimates

Consider a nonlinear backward parabolic problem in the form ut+Au(t)=f(t,u(t)),u(T)=g, where AA is a positive self-adjoint unbounded operator. Based on the fundamental solution to the parabolic equation, we propose to solve this problem by the Fourier truncated method, which generates a well-posed integral equation. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. Our regularizing scheme can be considered a new regularization, with the advantage of a relatively small amount of computation compared with the quasi-reversibility or quasi-boundary value regularizations. Error estimates for this method are provided together with a selection rule for the regularization parameter. These errors show that our method works effectively.