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 C0 semi-group on H. Let φ be in H, E > e a given positive number and let f : [0, T]×H → H satisfy the Lipschitz condition ∥f(t, w1)−f(t, w2)∥ ≤ k∥w1−w2∥,w1,w2∈H, for some non-negative constant k independent of t, w1 and w2. It is proved that if u1 and u2 are two solutions of the ill-posed semi-linear parabolic equation backward in time ut + Au = f(t, u), 0 < t ≤ T,∥u(T)−φ∥ ≤ e and ∥ui(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 Holder type.