Well-posedness for the backward problems in time for general time-fractional diffusion equation

In this article, we consider an evolution partial differential equation with Caputo time-derivative with the zero Dirichlet boundary condition: partial derivative(alpha)(t)u + Au - F where 0 < alpha < 1 and the principal part -A, is a non-symmetric elliptic operator of the second order. Given a source F. we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that -A is symmetric. The key is a perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator A.