Quasi-Reversibility Method for an Ill-Posed Nonhomogeneous Parabolic Problem

The quasi-reversibility method is considered for the non-homogeneous backward Cauchy problem ut+Au = f(t), u(τ) = ϕ for 0≤t<τ, which is known to be an ill-posed problem. Here, A is a densely defined positive self-adjoint unbounded operator on a Hilbert space H with given data f∈L1([0,τ],H) and ϕ∈H. Error analysis is considered when the data ϕ, f are exact and also when they are noisy. The results obtained generalize and simplify many of the results available in the literature.