Well-posedness for the backward problems in time for general time-fractional diffusion equation (Q2658506)

Summary: In this article, we consider an evolution partial differential equation with Caputo time-derivative with the zero Dirichlet boundary condition: \( \partial^{\alpha}_tu + Au = F\) 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 \(A\).