Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source

Let H be a Hilbert space with the inner product <., .> and the norm vertical bar vertical bar . vertical bar vertical bar, A(t) a positive self-adjoint unbounded time-dependent operator on H and epsilon > 0. We establish stability estimates of Holder type and propose a regularization method with error estimates of Holder type for the ill-posed backward semilinear parabolic equation {u(t)(t) + A(t)u(t) = f(t,u(t)), 0< t <= T, vertical bar vertical bar u(T) - phi vertical bar vertical bar <= epsilon, with the source function f satisfying a local Lipschitz condition.