Truncated spectral regularization for an ill-posed non-linear parabolic problem

It is known that the nonlinear nonhomogeneous backward Cauchy problem ut(t) + Au(t) = f (t, u(t)), 0 ⩽ t < τ with u(τ) = φ, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f, that a solution of the above problem satisfies an integral equation involving the spectral representation of A, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value φ. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.