Stability results for backward parabolic equations with time-dependent coefficients

Abstract

Let H be a Hilbert space with the norm ∥ ∥ and A(t) (0 ≤ t ≤ T ) be positive self-adjoint unbounded operators from D(A(t)) ⊂ H to H. In the paper, we establish stability estimates of Hölder type and propose a regularizationmethod for the ill-posed-backward parabolic equation with time-dependent coefficients {ut + A(t)u = 0, 0 < t < T{ ∥u(T ) - f ∥ ≤ ε, f ε H, ε > 0. Our stability estimates improve the related results by Krein (1957 Dokl. Akad. Nauk SSSR 114 1162-5), and Agmon and Nirenberg (1963 Commun. Pure Appl. Math. 16 121-239). Our regularization method with a priori and a posteriori parameter choice yields error estimates of Hölder type. This is the only result when a regularization method for backward parabolic equations with time-dependent coefficients provides a convergence rate. © 2011 IOP Publishing Ltd.