As we consider problem (1.1) in

with respect to variable

, we extend

, and other functions of variable

appearing in the paper to be zero for

. Throughout the paper, we assume that for the exact

the solution

exists and satisfies an apriori bound

where

is defined by

Since

is measured by the thermocouple, there will be measurement errors, and we would actually have as data some function

, for which

where the constant

represents a bound on the measurement error, and

denotes the

norm and

is the Fourier transform of function

. The problem (1.1) can be formulated, in frequency space, as follows:

Then we have the following lemma.

Lemma 2.1.

Problem (2.5)–(2.7) has the solution given by

where

denotes modified spherical Bessel function which given by [

21]

Proof.

Due to [

21], we can solve (2.5), in the frequency domain, to obtain

where

denotes also modified spherical Bessel function which is given by

Combining

with condition(2.7), we obtain

, that is,

According to [

21], there holds

where

, both

and

denote the Kelvin functions. Since

, we have

Therefore, for

,

Solving the systems (2.6) and (2.12) using (2.15) we get

Substitution of
in (2.16) into (2.12), we obtain (2.8).

Applying an inverse Fourier transform to (2.8), problem (1.1) has the solution

In order to obtain ill-posedness of problem (1.1) for
, we need the following lemma.

Lemma 2.2.

If function

satisfies (2.15), then there exist positive constants

such that, for

Proof.

First, due to [

21] and (2.15), we have, for

and

,

then there exist positive constants

such that, for

large enough, say

From these we know that there exist positive constants

and

such that, for

and

,

Then, since function

is continuous in the closed region

. Threrfore, there exist constants

and

such that, for

and

,

Finally, combining inequalities (2.22) with (2.23), we can see that there exist others constants
and
such that, for
, inequalities (2.18) are valid. Similarly, we obtain inequalities (2.19).

In order to formulate problem (1.1) for

in terms of an operator equation in the space

, we define an operator

, that is,

Denote

, and we can see that

is a multiplication operator:

From (2.26), we can prove the following lemma.

Lemma 2.3.

Let

be the adjoint to

, then

corresponds to the following problem where the left-hand side

of problem (1.1) is replaced by

, says

Proof.

Via the the following relations, combining with (2.26),

we can get the adjoint operator

of

in frequency domain

On the other hand, the problem (2.27) can be formulated, in frequency space, as follows:

Taking the conjugate operator for problem (2.5)–(2.7), we realize that

. Therefore, by Lemma 2.1, we conclude that

Hence the conclusion of Lemma 2.3 is proved.

The Parseval formula for the Fourier transform together with inequality (2.18), there holds

This implies that
, which is Fourier transform of exact data
, must decay rapidly at high frequencies since
. But such a decay is not likely to occur in the Fourier transform of the measured noisy data
at
. So, small perturbation of
in high frequency components can blow up and completely destroy the solution
given by (2.17) for
.