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

Abstract Consider a nonlinear backward parabolic problem in the form u t + A u ( t ) = f ( t , u ( t ) ) , u ( T ) = g , where A 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.

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