Acta Mathematica Vietnamica

Let $H$ be a Hilbert space with the norm $\|\cdot\|$, and let $A:D(A) \subset H \to H$ be a positive self-adjoint unbounded linear operator on H such that $−A$ generates a $C_0$ semi-group on $H$. Let $\varphi$ be in $H,$ $E > ε$ a given positive number and let $f : [0, T]\times H \to H$ satisfy the Lipschitz condition $\|f(t, w_1)−f(t, w_2)∥ \leq k\|w_1−w_2\|,$ $w_1,w_2\in H$, 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 $u_t + A u = f(t, u)$, $0 < t \leq T,$ $\|u(T)−\varphi\| \leq \varepsilon$ and $\|u_i(0)\| \leq E$, $i = 1,2,$ 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.

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