On the backward problem for parabolic equations with memory

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.

KEYWORDS:

• Backward parabolic equations
• memory effect
• regularization
• stability estimate

CLASSIFICATION AMS 2010:

• Primary 31B30
• 47A52
• 65F22
• 65J20

ORCID

Tra Quoc Khanh http://orcid.org/0000-0001-8643-777X

Disclosure statement

No potential conflict of interest was reported by the authors.