Bài báo khoa học

Let X be a Banach space with norm ‖ ⋅ ‖. Let A : D(A) ⊂ X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [0, T] → X satisfying ut + Au = 0, 0 < t < T, ‖u(T) − φ‖ ⩽ ε, for φ in X, is regularized by the generalized Sobolev equation uαt + Aαuα = 0, 0 < t < T, uα(T) = φ, where 0 < α < 1 and Aα = A(I + αAb)−1 with b ⩾ 1. Error estimates of the method with respect to the noise level are proved.