Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 99–111 | Cite as

Stability Results for Semi-linear Parabolic Equations Backward in Time

  • Nguyen Van DucEmail author
  • Nguyen Van Thang


Let H be a Hilbert space with the norm ∥⋅∥, and let A:D(A) ⊂ HH be a positive self-adjoint unbounded linear operator on H such that −A generates a C 0 semi-group on H. Let φ be in H, E > ε a given positive number and let f : [0, THH satisfy the Lipschitz condition ∥f(t, w 1)−f(t, w 2)∥ ≤ kw 1w 2∥,w 1,w 2H, for some non-negative constant k independent of t, w 1 and w 2. It is proved that if u 1 and u 2 are two solutions of the ill-posed semi-linear parabolic equation backward in time u t + A u = f(t, u), 0 < tT,∥u(T)−φ∥ ≤ ε and ∥u i (0)∥ ≤ E, i = 1,2, then
$$\|u_{1}(t)-u_{2}(t)\| \leq 2\varepsilon^{t/T} E^{1-t/T}\exp\Big[\Big(2k+\frac{1}{4}k^{2}(T+t)\Big)\frac{t(T-t)}{T}\Big] \quad \forall t \in [0,T]. $$
The ill-posed problem is stabilized by a modification of Tikhonov regularization which yields an error estimate of Hölder type.


Semi-linear parabolic equations backward in time Ill-posed problems Stability estimate Log-convexity method Tikhonov regularization 

Mathematics Subject Classification (2010)




This research was supported by Vietnam Ministry of Education and Training under grant number B2013-27-09.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsVinh UniversityVinh CityVietnam

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