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

Abstract

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