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On the backward problem for parabolic equations with memory

Authors: 

Tran Nhat Luana, Tra Quoc Khanh*

Source title: 
Applicable Analysis, 2019 (ISI)
Academic year of acceptance: 
2019-2020
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.