Stability Results for Semi-linear Parabolic Equations Backward in Time

Abstract

Let H be a Hilbert space with the norm ∥⋅∥, and let A:D(A) ⊂ H → H 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, T]×H → H satisfy the Lipschitz condition ∥f(t, w ₁)−f(t, w ₂)∥ ≤ k∥w ₁−w ₂∥,w ₁,w ₂∈H, 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 < t ≤ T,∥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.