Abstract
Let X be a Banach space with norm
∥
⋅
∥
{\|\cdot\|}
. Let
A
:
D
(
A
)
⊂
X
→
X
{A:D(A)\subset X\rightarrow X}
be an (possibly unbounded) operator that generates a
uniformly bounded holomorphic semigroup. Suppose that
ε
>
0
{\varepsilon>0}
and
T
>
0
{T>0}
are two given constants.
The backward parabolic equation of finding a function
u
:
[
0
,
T
]
→
X
{u:[0,T]\rightarrow X}
satisfying
u
t
+
A
u
=
0
,
0
<
t
<
T
,
∥
u
(
T
)
-
φ
∥
⩽
ε
,
u_{t}+Au=0,\quad 0<t<T,\;\|u(T)-\varphi\|\leqslant\varepsilon,
for φ in X, is regularized by the generalized Sobolev equation
u
α
t
+
A
α
u
α
=
0
,
0
<
t
<
T
,
u
α
(
T
)
=
φ
,
u_{\alpha t}+A_{\alpha}u_{\alpha}=0,\quad 0<t<T,\;u_{\alpha}(T)=\varphi,
where
0
<
α
<
1
{0<\alpha<1}
and
A
α
=
A
(
I
+
α
A
b
)
-
1
{A_{\alpha}=A(I+\alpha A^{b})^{-1}}
with
b
⩾
1
{b\geqslant 1}
. Error estimates of the method with respect to the noise level are proved.