A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
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We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates. Copyright © 2006 M. Denche and S. Djezzar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We consider the following final value problem (FVP) u (t) + Au(t) = 0,
0≤t0 be a spectral measure associated to the operator A in the Hilbert space H, then for all f ∈ H, we can write f=
∞ 0
dEλ f .
(2.1)
If the (FVP) problem (resp., (QBVP) problem) admits a solution u (resp., uα ), then this solution can be represented by u(t) =
∞ 0
eλ(T −t) dEλ f ,
(2.2)
e−λt dEλ f . αλ + e−λT
(2.3)
respectively, uα (t) =
∞ 0
Theorem 2.2. For all f ∈ H, the functions uα given by (2.3) are classical solutions to the (QBVP) problem and we have the following estimate uα (t) ≤
T f , α 1 + ln(T/α)
where α < eT.
∀t ∈ [0,T],
(2.4)
M. Denche and S. Djezzar
3
Proof. If we assume that the functions uα given in (2.3) are defined for all t ∈ [0,T], then, it is easy to show that uα ∈ C 1 ([0,T],H) and ∞
uα (t) =
−λe−λt
αλ + e−λT
0
dEλ f .
(2.5)
From Auα (t)2 =
∞ 0
2 λ2 e−2λt 1 2 d Eλ f ≤ 2 − λT α αλ + e
∞
0
2
dEλ f =
1 f 2 , α2
(2.6)
we get uα (t) ∈ D(A) and so uα ∈ C([0,T],D(A)). This shows that the function uα is a classical solution to the (QBVP) problem. Now, using (2.3), we have uα (t)2 ≤
∞
2 1 2 d Eλ f , − λT αλ + e
0
(2.7)
if we put
h(λ) = αλ + e−λT
−1
,
for λ > 0,
(2.8)
then,
sup h(λ) = h λ>0
ln(T/α) , T
(2.9)
and this yields uα (t)2 ≤
T α 1 + ln(T/α)
2 ∞ 0
2
dEλ f =
T α 1 + ln(T/α)
2 f 2 .
(2.10)
This shows that the integral defining uα (t) exists for all t ∈ [0,T] and we have the desired estimate. Remark 2.3. One advantage of this method of regularization is that the order of the error, introduced by small changes in the final value f , is less than the order given in [4]. Now, we give the following convergence result. Theorem 2.4. For every f ∈ H, uα (T) converges to f in H, as α tends to zero. Proof. Let ε > 0, choose η > 0 for which ∞ η
2 ε dEλ f < . 2
(2.11)
From (2.3), we have uα (T) − f 2 ≤ α2
η 0
2 ε λ2 2 d Eλ f + , − λT 2 αλ + e
(2.12)
4
Regularization of parabolic ill-posed problems
so by choosing α such that η
α2 < ε 2
0
2
λ2 e2λT Eλ f
−1
,
(2.13)
we obtain the desired r
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