Uniqueness of the potential in a time-fractional diffusion equation

A Proof of Lemma 3.6

By using the eigenpairs { ( λ n ⁢ ( q ) , ϕ n ⁢ ( x ; q ) ) } , the Green function G ⁢ ( x , y ; q ) for the differential operator L ⁢ ( q ) and the Mittag-Leffler function E α , β ⁢ ( z ) , the unique solution u to equation (1.1) with source term f ∈ L 2 ⁢ ( 0 , 1 ) can be represented as follows, where u ⁢ ( x , t ) ∈ C ⁢ ( [ 0 , T ] ; H 2 ⁢ ( 0 , 1 ) ) (see [20, Theorem 2.2, Theorem 2.4]):

where the second line follows from Lemma 3.3, and the third line from the definition of the Green function.

Let us first define the following function:

It follows from the solution representation of (1.1) that

According to the completeness of the set { f j } in L 2 ⁢ ( 0 , 1 ) , it is straightforward to show that

Set v 1 = G ⁢ ( 1 , y ; q 1 ) and v 2 = G ⁢ ( 1 , y ; q 2 ) . From Lemma 3.4, we have that v = v 1 - v 2 satisfies

Therefore, through the integration of the first line above and (A.1), for all η ∈ C c ∞ ⁢ ( 0 , 1 ) , we obtain

By integration by parts and the properties of the eigenfunctions ϕ n ⁢ ( y ; q ) , there holds

We rewrite the summand ζ in I 1 as follows:

Next, we need to estimate the terms ζ 1 , ζ 2 , ζ 3 . Combining Lemmas 3.1–3.3 and the properties of eigenpairs { λ n , ϕ n } n ∈ ℕ (see [3]), we infer that

where λ ~ n lies within λ n ⁢ ( q 1 ) and λ n ⁢ ( q 2 ) . Therefore

Similarly, some calculations give

and

Moreover,

Through the estimate above and (3.3)–(3.4), it follows that

Applying the estimate above, we finally have

The proof of (b) is completed. The proof of part (a) is nearly identical and hence omitted.