ABSTRACT
We study the backward problem for the parabolic equation with memory. We prove that the nonlocal problem is well-posed for 0<t<T and is ill-posed only at the initial time of t=0. This result makes a big difference between the classical backward parabolic equation and the backward parabolic equation with memory. We further propose a spectral regularization method to overcome the ill-posedness at the initial time. The Hölder convergence rate is obtained under both a priori and a posteriori parameter choice rules.
CLASSIFICATION AMS 2010:
ORCID
Tra Quoc Khanh http://orcid.org/0000-0001-8643-777X
Disclosure statement
No potential conflict of interest was reported by the authors.