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^{\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 .

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