Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Wenjun Ma; Liangliang Sun

doi:10.1515/cmam-2022-0210

Publicly Available Published by De Gruyter July 12, 2023

Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Wenjun Ma and Liangliang Sun

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2022-0210

Abstract

This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.

Keywords: Multi-Term Time-Fractional Diffusion Equation; Multi-Parameter Inversion; Potential Term; Existence and Uniqueness; Stability

MSC 2010: 35R30; 35R25; 65M30

1 Introduction

Let Ω be a bounded domain in R d for d ≤ 3 with sufficiently smooth boundary ∂ Ω . For a fixed positive integer 𝑚, the orders α = ( α 1 , … , α m ) and the coefficients q = ( q 1 , … , q m ) are restricted in the admissible sets

A := { ( α 1 , … , α m ) ∈ R m ; α ̄ ≥ α 1 > α 2 > ⋯ > α m ≥ α ¯ } , Q := { ( q 1 , … , q m ) ∈ R m ; q 1 = 1 , q j ∈ [ q ¯ , q ̄ ] ⁢ ( j = 2 , … , m ) }

with fixed 0 < α ¯ < α ̄ < 1 and 0 < q ¯ < q ̄ . Consider the following initial boundary value problem (IBVP) for a multi-term time-fractional diffusion equation:

(1.1) { ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x , t ) + L ⁢ u ⁢ ( x , t ) − c ⁢ ( t ) ⁢ u ⁢ ( x , t ) = r ⁢ ( x ) ⁢ p ⁢ ( t ) , x ∈ Ω , 0 < t ≤ T , B σ ⁢ u ⁢ ( x , t ) = 0 , x ∈ ∂ Ω , 0 < t ≤ T , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) , x ∈ Ω ,

where ∂ t α denotes the Caputo fractional left-sided derivative defined by

∂ t α u ⁢ ( x , t ) = 1 Γ ⁢ ( 1 − α ) ⁢ ∫ 0 t ∂ u ⁢ ( x , s ) ∂ s ⁢ d ⁢ s ( t − s ) α , t > 0 ,

in which Γ ⁢ ( ⋅ ) is the Gamma function (see Kilbas, Srivastava and Trujillo [20] and Podlubny [39]).

The differential operator 𝐿 is defined by

L ⁢ u ⁢ ( x , t ) = − ∑ i , j = 1 d ∂ ∂ x i ⁢ ( a i ⁢ j ⁢ ( x ) ⁢ ∂ u ∂ x j ⁢ ( x , t ) ) + b ⁢ ( x ) ⁢ u ⁢ ( x , t ) ,

where a i ⁢ j = a j ⁢ i , 1 ≤ i , j ≤ d . Moreover, we assume that the operator 𝐿 is uniformly elliptic on Ω ̄ , i.e. there exists a constant μ > 0 such that

∑ i , j = 1 d a i ⁢ j ⁢ ( x ) ⁢ ξ i ⁢ ξ j ≥ μ ⁢ | ξ | 2 , x ∈ Ω ̄ , ξ = ( ξ 1 , … , ξ d ) ∈ R d .

The boundary condition B σ is defined as

B σ ⁢ u ⁢ ( x ) = ( 1 − σ ⁢ ( x ) ) ⁢ u ⁢ ( x ) + σ ⁢ ( x ) ⁢ ∂ ν u ⁢ ( x ) , x ∈ ∂ Ω ,

where

∂ ν u ⁢ ( x ) = ∑ i , j = 1 d a i ⁢ j ⁢ ( x ) ⁢ ∂ u ∂ x i ⁢ ν j ⁢ ( x )

and ν = ( ν 1 , … , ν d ) is the outward normal unit vector to ∂ Ω . Here σ ⁢ ( x ) ∈ C 2 ⁢ ( ∂ Ω ) satisfying 0 ≤ σ ⁢ ( x ) ≤ 1 . The regularity for a i ⁢ j depends on whether σ ≡ 0 or not, i.e.

a i ⁢ j ∈ { C 1 ⁢ ( Ω ̄ ) if ⁢ σ ≡ 0 , C 2 ⁢ ( Ω ̄ ) if ⁢ σ ≢ 0 ,

and b ⁢ ( x ) ∈ C ⁢ ( Ω ̄ ) , b ⁢ ( x ) ≥ 0 if σ = 0 , or b ⁢ ( x ) > 0 if σ = 1 .

Throughout this paper, we always assume the following conditions hold:

(1.2) c ⁢ ( t ) ∈ L ∞ ⁢ ( 0 , T ) ,

(1.3) p ⁢ ( t ) ∈ L ∞ ⁢ ( 0 , T ) ,

(1.4) r ⁢ ( x ) ∈ H 2 ⁢ ( Ω ) ⁢ and ⁢ B σ ⁢ r = 0 on ⁢ ∂ Ω ,

(1.5) u 0 ⁢ ( x ) ∈ H 4 ⁢ ( Ω ) ⁢ and ⁢ B σ ⁢ u 0 = B σ ⁢ ( L ⁢ u 0 ) = 0 on ⁢ ∂ Ω .

As we know, the time-fractional diffusion models like (1.1) in the case of m = 1 (called single-term counterpart) deduced by replacing the standard time derivative with a time-fractional derivative have received great attention in applied disciplines, e.g., the slower diffusion phenomenon, anomalous non-Fickian growth rates, skewness and long-tailed profile in a highly heterogeneous aquifer [1, 3, 23] and anomalous diffusion models derived from continuous time random walk at the macro level [33], and so on. At the same time, several researchers also found which orders of the derivatives change in time and/or spatial coordinates. One important particular case of fractional diffusion equations of changed order is the multi-term time-fractional diffusion equation just like (1.1), whose mean-squared displacement behaving like ⟨ Δ ⁢ x 2 ⟩ ∼ C ⁢ t min ⁡ { α j } as t → ∞ (see, e.g., [31]) grows in time slower than linearly. This kind of model could efficiently describe the diffusion phenomenon of a solute in a multi-scale medium, such as modeling a crowd of non-homogeneous and non-stationary processes. For example, it is shown to be an efficient model for describing some anomalous diffusion processes in the highly heterogeneous media by [11] in which the authors indicated that a diffusion equation with time-fractional derivative performs well in describing the long-tailed profile of a particle diffuses in a highly heterogeneous medium. Also, the multi-term model (1.1) employs multiple fractional orders to improve the modeling accuracy of the single-term models. For example, it is demonstrated in [41] that a two-term fractional-order diffusion model was proposed for the total concentration in solute transport, in order to describe the mobile and immobile status of the solute. The model with two fractional derivatives appears also naturally when describing sub-diffusive motion in velocity fields [34].

Due to unprecedented modeling capability of model (1.1), the analytical study of the direct problem of (1.1) has received much more attention in recent years, including maximum principle, well-posedness, long-time asymptotic behavior and analytic continuation; please refer to [29, 40, 26, 27, 28, 8, 21, 19]. Some numerical approaches, such as finite difference method, finite element method and spectral method, and so on, are developed in the papers [24, 55, 51, 15, 42, 5].

Although the research on the time-fractional diffusion equations has achieved fruitful results; however, some important parameters, such as the number of fractional derivatives 𝑚, orders 𝜶 or its coefficients 𝒒, diffusion coefficient a i ⁢ j ⁢ ( x ) , potential term c ⁢ ( t ) or b ⁢ ( x ) , source term p ⁢ ( t ) or r ⁢ ( x ) , initial state u 0 ⁢ ( x ) or the boundary condition coefficient σ ⁢ ( x ) , are always unknown in practice. This gives rise to the inverse problems for time-fractional diffusion equations. In the present paper, the time-dependent potential term c ⁢ ( t ) and time source term p ⁢ ( t ) are unknown, so we need to consider the following inverse problems.

Assume σ ≡ 0 , i.e. B σ ⁢ u ⁢ ( x , t ) = u ⁢ ( x , t ) = 0 on ∂ Ω , and let x 1 , x 2 ∈ Ω ( x 1 ≠ x 2 ). Given all the definite solution conditions except for unknown potential term c ⁢ ( t ) and time source p ⁢ ( t ) , can we find a pair ( c ⁢ ( t ) , p ⁢ ( t ) ) uniquely such that the solution 𝑢 for IBVP (1.1) satisfies
u ⁢ ( x i , t ) = h i D ⁢ ( t ) , 0 < t < T , i = 1 , 2 ,
where h i D ⁢ ( t ) is the measured data given in advance?
Assume σ ≢ 0 , i.e. B σ ⁢ u ⁢ ( x , t ) = ( 1 − σ ⁢ ( x ) ) ⁢ u ⁢ ( x , t ) + σ ⁢ ( x ) ⁢ ∂ ν u ⁢ ( x , t ) = 0 on ∂ Ω , and let x 1 , x 2 ∈ Ω ̄ ( x 1 ≠ x 2 ). Given all the definite solution conditions except for unknown potential term c ⁢ ( t ) and time source p ⁢ ( t ) , can we find a pair ( c ⁢ ( t ) , p ⁢ ( t ) ) uniquely such that the solution 𝑢 for IBVP (1.1) satisfies
u ⁢ ( x i , t ) = h i R ⁢ ( t ) , 0 < t < T , i = 1 , 2 ,
where h i R ⁢ ( t ) is the measured data given in advance?

The above problem is a multi-parameter simultaneous inversion problem. So there are some difficulties and challenges including theory analysis and numerical implementation, such as uniqueness and conditional stability of the inverse problem, regularization method and reconstruction algorithm, and so on. For the single parameter determination of potential term, we refer to the papers [16, 49, 35, 46, 44, 54, 52, 53, 18, 14, 48, 13, 47]. Jin and Rundell [16] obtained a uniqueness result in determining the potential term from the flux measurements in one dimension with the Dirichlet boundary conditions. Yamamoto and Zhang [49] gave a conditional stability estimate in determining the potential in a half-order fractional diffusion equation by a Carleman estimate. Miller and Yamamoto [35] discussed an inverse problem of determining the spatially dependent potential and fractional order from the internal data. Sun, Zhang and Wei [44] investigated the stability of an inverse time-dependent potential in a multi-term time-fractional diffusion equation and proposed a Levenberg–Marquardt method to reconstruct the potential. Zhang and Zhou [54] studied a recovering potential term in one-dimensional space from the final time data and gave the uniqueness of inverse problem and an efficient algorithm. Also, there are some studies on simultaneously determining potential and other parameters, e.g., in the papers [25, 43, 45, 50, 17].

In this paper, we focus on simultaneous inversion of the time-dependent potential term c ⁢ ( t ) and time source p ⁢ ( t ) by two-point measurements u ⁢ ( x i , t ) , x i ∈ Ω ̄ , i = 1 , 2 . To the authors’ best knowledge, there are no works on multi-parameter inversion of time-dependent potential and time source problem at present for a multi-term time-fractional diffusion equation. In this study, we obtain the existence, uniqueness and some regularities of the solution for the direct problem and also obtain a nice stability of the inverse problem. Moreover, we employ the modified Levenberg–Marquardt method to solve numerically the inverse problem. The numerical results for three examples are provided to show the effectiveness and robustness of the proposed methods.

The main contribution of the paper lies in the conditional stability of the inverse problem and an effective reconstruction algorithm of multi-parameter inversion based on the Levenberg–Marquardt method. The results of this paper are an extension of the previous paper [44]. First we generalize the single-parameter inversion to a multi-parameter simultaneous inversion problem and also obtain a very good conditional stability estimate of the inverse problem (e.g., see (4.3) of Theorem 4.3), and secondly, the theoretical results for forward and inverse problems are also extended from a single-term fractional equation to a multi-term case. Compared with the single-parameter inversion problem in literature [44], since we need to further estimate the regularity of the solution of the direct problem in order to obtain a nice conditional stability of the inverse problem in this paper, it is necessary to redefine the workspace in the application of the fixed point theorem, which makes the proof process more complicated than the single case. On the other hand, in the theoretical proof of the inverse problem, multi-parameter inversion involves the solving of linear equations compared with the single-term case. Therefore, a condition (see (4.1) of Theorem 4.3) on the solution information at the measurement point is needed, which also provides a theoretical basis for the selection of measurement points for the numerical reconstruction of the inverse problem. Finally, we modify the Levenberg–Marquardt method and apply it to multi-parameter inversion. We have well reconstructed two unknown parameters by introducing two regularization parameters. And it also can be applied to other inverse problems. Although similar results have been studied in other literature (see, e.g., [50, 47]), however, it is a pity that the authors ignored the regularity of the solution to the corresponding forward problem when proving the conditional stability of the inverse problem.

The remainder of this paper is organized as follows. Some preliminaries are presented in Section 2. In Section 3, we give the existence, uniqueness and some regularities of the solution for the direct problem. In Section 4, we give the conditional stability result of inverse problem. In Section 5, we illustrate the ill-posedness of inverse problem and use the modified Levenberg–Marquardt method to find the approximations of time-dependent potential and time source. Numerical results for three examples are provided to illustrate the efficiency of our method in Section 6. Finally, we give a brief conclusion in Section 7.

2 Preliminaries

Denote ∥ ⋅ ∥ = ∥ ⋅ ∥ L 2 ⁢ ( Ω ) , ∥ ⋅ ∥ ∞ = ∥ ⋅ ∥ L ∞ ⁢ ( 0 , T ) , ( ⋅ , ⋅ ) as the inner product of L 2 ⁢ ( Ω ) and H s ⁢ ( Ω ) , s ∈ R , is the Sobolev space (see Adams [2]).

Let

D ⁢ ( L ) := { u ∈ H 2 ⁢ ( Ω ) ; B σ ⁢ u = 0 ⁢ on ⁢ ∂ Ω }

denote the domain of the operator 𝐿. Noting that 𝐿 is a self-adjoint and positive operator. Let { λ k , ϕ k } k = 1 ∞ be an eigensystem of 𝐿 in D ⁢ ( L ) . Then we know 0 < λ 1 < λ 2 ≤ λ 3 ≤ ⋯ , lim k → ∞ λ k = ∞ , L ⁢ ϕ k = λ k ⁢ ϕ k , and { ϕ k } k = 1 ∞ ⊂ H 2 ⁢ ( Ω ) forms an orthonormal basis of L 2 ⁢ ( Ω ) . We can define the Hilbert scale space D ⁢ ( L γ ) for γ ≥ 0 (see, e.g., [38]) by

D ⁢ ( L γ ) = { ψ ∈ L 2 ⁢ ( Ω ) ; ∑ k = 1 ∞ λ k 2 ⁢ γ ⁢ | ( ψ , ϕ k ) | 2 < ∞ } , L γ ⁢ ψ = ∑ k = 1 ∞ λ k γ ⁢ ( ψ , ϕ k ) ⁢ ϕ k , ψ ∈ D ⁢ ( L γ ) ,

equipped with the norm ∥ ψ ∥ D ⁢ ( L γ ) = ∥ L γ ⁢ ψ ∥ . According to [7, 9], we have

(2.1) D ⁢ ( L γ ) ⊂ H 2 ⁢ γ ⁢ ( Ω ) , 0 ≤ γ ≤ 1 ,

(2.2) C 1 ⁢ ∥ ψ ∥ H 2 ⁢ γ ⁢ ( Ω ) ≤ ∥ ψ ∥ D ⁢ ( L γ ) ≤ C 2 ⁢ ∥ ψ ∥ H 2 ⁢ γ ⁢ ( Ω ) , ψ ∈ D ⁢ ( L γ ) , 0 ≤ γ ≤ 1 , γ ≠ 1 4 .

Definition 2.1

The multinomial Mittag-Leffler function (see [30, 20]) is defined by

E ( θ 1 , … , θ m ) , θ 0 ⁢ ( z 1 , … , z m ) := ∑ k = 0 ∞ ∑ k 1 + ⋯ + k m = k ( k ; k 1 , … , k m ) ⁢ ∏ j = 1 m z j k j Γ ⁢ ( θ 0 + ∑ j = 1 m θ j ⁢ k j ) ,

where θ 0 , θ j ∈ R , and z j ∈ C ( j = 1 , … , m ), and ( k ; k 1 , … , k m ) denotes the multinomial coefficient

( k ; k 1 , … , k m ) := k ! k 1 ! ⁢ ⋯ ⁢ k m ! with ⁢ k = ∑ j = 1 m k j .

For later use, if orders α = ( α 1 , … , α m ) and coefficients q = ( q 1 , … , q m ) are restricted in the admissible sets 𝐀 and 𝐐 defined in the above text, respectively, we adopt the abbreviation

(2.3) E α ′ , β ( n ) ⁢ ( t ) := E ( α 1 , α 1 − α 2 , … , α 1 − α m ) , β ⁢ ( − λ n ⁢ t α 1 , − q 2 ⁢ t α 1 − α 2 , … , − q m ⁢ t α 1 − α m ) , t > 0 ,

where λ n denotes the 𝑛th eigenvalue of the elliptic operator 𝐿 in D ⁢ ( L ) .

Proposition 2.2

Proposition 2.2 ([26])

Let β > 0 and 1 > α 1 > ⋯ > α m > 0 be given. Assume α 1 ⁢ π / 2 < μ < α 1 ⁢ π , μ ≤ | arg ⁡ ( z 1 ) | ≤ π , and z j ∈ R for j = 2 , … , m , and there exists K > 0 such that − K ≤ z j < 0 ( j = 2 , … , m ). Then there exists a constant C > 0 depending only 𝜇, 𝐾, α j ( j = 1 , … , m ) and 𝛽 such that

| E ( α 1 , α 1 − α 2 , … , α 1 − α m ) , β ⁢ ( z 1 , … , z m ) | ≤ C 1 + | z 1 | .

Lemma 2.3

Lemma 2.3 ([4])

Let f ∈ L p ⁢ ( 0 , T ) and g ∈ L q ⁢ ( 0 , T ) with 1 ≤ p , q ≤ ∞ and 1 / p + 1 / q = 1 . Then the function f ∗ g defined by f ∗ g ⁢ ( t ) = ∫ 0 t f ⁢ ( t − s ) ⁢ g ⁢ ( s ) ⁢ d s belongs to C ⁢ [ 0 , T ] and satisfies

| f ∗ g ⁢ ( t ) | ≤ ∥ f ∥ L p ⁢ ( 0 , t ) ⁢ ∥ g ∥ L q ⁢ ( 0 , t ) , t ∈ [ 0 , T ] .

Lemma 2.4

Lemma 2.4 (see [47, Lemma 2.7])

Let 0 < α < 1 . Suppose f ∈ W 1 , ∞ ⁢ ( 0 , T ) and ∥ f ′ ∥ L ∞ ⁢ ( 0 , T ) ≤ E ; then we have

∥ ∂ t α f ∥ C ⁢ [ 0 , T ] ≤ C ⁢ E α ⁢ ∥ f ∥ C ⁢ [ 0 , T ] 1 − α ,

where C = C ⁢ ( α ) > 0 is independent of 𝑓.

3 Existence, Uniqueness and Regularity of the Solution for the Direct Problem

In this section, we will obtain the existence, uniqueness and regularity of the solution for direct problem (1.1).

Here we firstly introduce Banach spaces C 0 ⁢ [ 0 , T ] and C 0 ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) defined by

{ u ∈ C ⁢ [ 0 , T ] ; u ⁢ has a compact support in ⁢ ( 0 , T ] , i.e. ⁢ u ⁢ ( 0 ) = 0 } , { u ∈ C ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) ; ∥ u ⁢ ( t ) ∥ ⁢ has a compact support in ⁢ ( 0 , T ] , i.e. ⁢ ∥ u ⁢ ( 0 ) ∥ = 0 } .

Theorem 3.1

Let conditions (1.2)–(1.5) hold. Then IBVP (1.1) has a unique solution u ∈ C ⁢ ( [ 0 , T ] ; H 2 + 2 ⁢ γ ⁢ ( Ω ) ) satisfying

L ⁢ u ∈ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) and ∂ t α j u ∈ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) , j = 1 , 2 , … , m ,

for 0 ≤ γ < 1 . Moreover, we have the following estimate:

(3.1) ∥ L ⁢ u ∥ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) + ∥ ∑ j = 1 m q j ⁢ ∂ t α j u ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C ⁢ ( ∥ p ∥ ∞ ⁢ ∥ r ∥ H 2 ⁢ ( Ω ) + ∥ u 0 ∥ H 4 ⁢ ( Ω ) )

with C > 0 depending on α 1 , Ω, 𝑇, 𝛾 and ∥ c ∥ ∞ .

Moreover, if p ∈ W 1 , ∞ ⁢ ( 0 , T ) ∩ C 0 ⁢ [ 0 , T ] and c ∈ W 1 , ∞ ⁢ ( 0 , T ) , then ∂ t u ∈ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) with 0 ≤ γ < 1 , and

(3.2) ∥ ∂ t u ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ C ⁢ ( ∥ p ∥ W 1 , ∞ ⁢ ( 0 , T ) ⁢ ∥ r ∥ + ∥ u 0 ∥ H 2 ⁢ ( Ω ) ) ,

where C > 0 depends on α 1 , Ω, 𝑇, 𝛾 and ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) .

In order to obtain the above results, we firstly study the well-posedness for the following IBVP with a general source:

(3.3) { ∑ j = 1 m q j ⁢ ∂ t α j v ⁢ ( x , t ) + L ⁢ v ⁢ ( x , t ) − c ⁢ ( t ) ⁢ v ⁢ ( x , t ) = f ⁢ ( x , t ) , x ∈ Ω , 0 < t ≤ T , B σ ⁢ v ⁢ ( x , t ) = 0 , x ∈ ∂ Ω , 0 < t ≤ T , v ⁢ ( x , 0 ) = 0 , x ∈ Ω .

Here we assume

(3.4) f ⁢ ( x , t ) ∈ L ∞ ⁢ ( 0 , T ; D ⁢ ( L ) ) .

By the fixed point theorem, we can obtain the following existence, uniqueness and regularities of the solution to problem (3.3).

Lemma 3.2

Assume 0 ≤ γ < 1 . Let (1.2) and (3.4) hold. Then IBVP (3.3) has a unique solution v ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) satisfying

∑ j = 1 m q j ⁢ ∂ t α j v ∈ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) )

and

(3.5) ∥ L ⁢ v ∥ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) + ∥ ∑ j = 1 m q j ⁢ ∂ t α j v ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C 1 ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) )

with C 1 > 0 depending on α 1 , Ω, 𝑇, 𝛾 and ∥ c ∥ L ∞ ⁢ ( 0 , T ) .

Moreover, if f ∈ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ∩ C 0 ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) and c ∈ W 1 , ∞ ⁢ ( 0 , T ) , then ∂ t v ∈ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) , and

(3.6) ∥ ∂ t v ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ C 2 ⁢ ∥ f ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ,

where C 2 > 0 depends on α 1 , Ω, 𝑇, 𝛾 and ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) .

Proof

Firstly, we define the operator-valued function K ⁢ ( t ) by

K ⁢ ( t ) ⁢ ψ = ∑ k = 1 ∞ ( ψ , ϕ k ) ⁢ t α 1 − 1 ⁢ E α ′ , α 1 ( k ) ⁢ ( t ) ⁢ ϕ k , ψ ∈ L 2 ⁢ ( Ω ) , t > 0 ,

where E α ′ , α 1 ( k ) ⁢ ( t ) is defined by (2.3).

From L γ ⁢ K ⁢ ( t ) ⁢ ψ = ∑ k = 1 ∞ λ k γ ⁢ ( ψ , ϕ k ) ⁢ t α 1 − 1 ⁢ E α ′ , α 1 ( k ) ⁢ ( t ) ⁢ ϕ k and Proposition 2.2, we have for 0 ≤ γ < 1 that

(3.7) ∥ L γ ⁢ K ⁢ ( t ) ⁢ ψ ∥ = ( ∑ k = 1 ∞ [ λ k γ ⁢ ( ψ , ϕ k ) ⁢ t α 1 − 1 ⁢ E α ′ , α 1 ( k ) ⁢ ( t ) ] 2 ) 1 / 2 ≤ C ⁢ ( ∑ k = 1 ∞ ( λ k γ ⁢ t α 1 − 1 1 + λ k ⁢ t α 1 ⁢ ( ψ , ϕ k ) ) 2 ) 1 / 2 = t α 1 − 1 − α 1 ⁢ γ ⁢ ( ∑ k = 1 ∞ ( ( λ k ⁢ t α 1 ) γ 1 + λ k ⁢ t α 1 ⁢ ( ψ , ϕ k ) ) 2 ) 1 / 2 ≤ C ⁢ t α 1 ⁢ ( 1 − γ ) − 1 ⁢ ∥ ψ ∥ , ψ ∈ L 2 ⁢ ( Ω ) , t > 0 ,

where C > 0 depends on Ω, α 1 , 𝛾, and here we use a fact that ( λ k ⁢ t α 1 ) γ 1 + λ k ⁢ t α 1 is bounded for each k ∈ N , γ ≤ 1 and t ≥ 0 .

Now we consider the following Cauchy problems in L 2 ⁢ ( Ω ) :

(3.8) { ∑ j = 1 m q j ⁢ ∂ t α j ω ⁢ ( t ) + L ⁢ ω ⁢ ( t ) = F ⁢ ( t ) , t ∈ ( 0 , T ] , ω ⁢ ( 0 ) = 0 ,

By [26, Theorem 2.2], for F ∈ L p ⁢ ( 0 , T ; D ⁢ ( L γ ) ) , p ∈ [ 1 , ∞ ] , γ ∈ [ 0 , 1 ] , we know that (3.8) admits a unique solution given by

(3.9) ω ⁢ ( t ) = ∫ 0 t K ⁢ ( t − s ) ⁢ F ⁢ ( s ) ⁢ d s .

On the other hand, IBVP (3.3) could be written as

(3.10) { ∑ j = 1 m q j ⁢ ∂ t α j v ⁢ ( t ) + L ⁢ v ⁢ ( t ) = c ⁢ ( t ) ⁢ v ⁢ ( t ) + f ⁢ ( t ) , t ∈ ( 0 , T ] , v ⁢ ( 0 ) = 0 ,

where v ⁢ ( t ) = v ⁢ ( ⋅ , t ) and f ⁢ ( t ) = f ⁢ ( ⋅ , t ) . Therefore, we see from (3.9) that the solution 𝑣 of (3.10) can be written as an integral equation in the following form:

(3.11) v ( t ) = ∫ 0 t K ( t − s ) f ( s ) d s + ∫ 0 t K ( t − s ) ( c ( s ) v ( s ) ) d s = : ( Q f ) ( t ) + ( Q c v ) ( t ) .

In addition, we obtain for f ∈ L p ( 0 , T ; D ( L ) that

L 1 + γ ⁢ Q ⁢ f ⁢ ( t ) = ∫ 0 t L γ ⁢ K ⁢ ( t − s ) ⁢ L ⁢ f ⁢ ( s ) ⁢ d s .

(3.12) ∥ L ⁢ Q ⁢ f ⁢ ( t ) ∥ D ⁢ ( L γ ) = ∥ L 1 + γ ⁢ Q ⁢ f ⁢ ( t ) ∥ ≤ ∫ 0 t ∥ L γ ⁢ K ⁢ ( t − s ) ⁢ ( L ⁢ f ) ⁢ ( s ) ∥ ⁢ d s ( by the generalized Minkowski inequality ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ∥ f ⁢ ( s ) ∥ D ⁢ ( L ) ⁢ d s ( by ( 3.7 ) ) ≤ C ⁢ T α 1 ⁢ ( 1 − γ ) − 1 / p ⁢ ∥ f ∥ L p ⁢ ( 0 , T ; D ⁢ ( L ) ) ( by Lemma 2.3 ) .

By (3.7), the map t ↦ L γ ⁢ K ⁢ ( t ) belongs to L q ⁢ ( 0 , T ; B ⁢ ( L 2 ⁢ ( Ω ) ) ) for q < 1 / ( 1 − α 1 ⁢ ( 1 − γ ) ) , where B ⁢ ( L 2 ⁢ ( Ω ) ) denotes a space of bounded linear operators in L 2 ⁢ ( Ω ) . Thus, by Lemma 2.3, for p > 1 / ( α 1 ⁢ ( 1 − γ ) ) satisfying 1 / p + 1 / q = 1 , we have Q ⁢ f ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L γ + 1 ) ) and

(3.13) ∥ Q ⁢ f ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ C ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L ) ) ,

where C > 0 depends on α 1 , Ω, 𝑇 and 𝛾.

On the other hand, from (3.11), we know

Q ⁢ f ⁢ ( t ) = ∫ 0 t K ⁢ ( t − s ) ⁢ f ⁢ ( s ) ⁢ d s = ∫ 0 t K ⁢ ( s ) ⁢ f ⁢ ( t − s ) ⁢ d s = ∑ k = 1 ∞ ∫ 0 t ( f ⁢ ( t − s ) , ϕ k ) ⁢ s α 1 − 1 ⁢ E α ′ , α 1 ( k ) ⁢ ( s ) ⁢ d s ⁢ ϕ k .

From f ∈ C 0 ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) , we have ∥ f ⁢ ( 0 ) ∥ = 0 . That means ( f ⁢ ( 0 ) , ϕ k ) = 0 for each k ∈ N + . Therefore, we obtain

d d ⁢ t ⁢ Q ⁢ f ⁢ ( t ) = K ⁢ ( t ) ⁢ f ⁢ ( 0 ) + ∫ 0 t K ⁢ ( s ) ⁢ f t ⁢ ( t − s ) ⁢ d ⁢ s = K ⁢ ( t ) ⁢ f ⁢ ( 0 ) + ∫ 0 t K ⁢ ( t − s ) ⁢ f s ⁢ ( s ) ⁢ d ⁢ s = ∑ k = 1 ∞ ( f ⁢ ( 0 ) , ϕ k ) ⁢ t α 1 − 1 ⁢ E α ′ , α 1 ( k ) ⁢ ( t ) ⁢ ϕ k + ∫ 0 t K ⁢ ( t − s ) ⁢ f s ⁢ ( s ) ⁢ d s = ∫ 0 t K ⁢ ( t − s ) ⁢ f s ⁢ ( s ) ⁢ d s .

Then we arrive at

(3.14) ∥ d d ⁢ t ⁢ Q ⁢ f ⁢ ( t ) ∥ D ⁢ ( L γ ) ≤ ∫ 0 t ∥ L γ ⁢ K ⁢ ( t − s ) ⁢ f s ⁢ ( s ) ∥ ⁢ d s ( by the generalized Minkowski inequality ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ∥ f s ⁢ ( s ) ∥ ⁢ d s ( by ( 3.7 ) ) ≤ C ⁢ T α 1 ⁢ ( 1 − γ ) ⁢ ∥ f ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ( by Lemma 2.3 ) .

Therefore, similar to the discussion below (3.12), we obtain d d ⁢ t ⁢ Q ⁢ f ⁢ ( t ) ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L γ ) ) , and

(3.15) ∥ d d ⁢ t ⁢ Q ⁢ f ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L γ ) ) ≤ C ⁢ ∥ f ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ,

where C > 0 depends on α 1 , Ω, 𝑇 and 𝛾.

Now we define X = C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ∩ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) equipped with the norm

∥ ⋅ ∥ X = ∥ ⋅ ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) + ∥ ⋅ ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) .

This can easily verify that 𝑋 is a Banach space. For each v ∈ X , define Q c ⁢ v by (3.11), namely

Q c ⁢ v ⁢ ( t ) = ∫ 0 t K ⁢ ( t − s ) ⁢ ( c ⁢ ( s ) ⁢ v ⁢ ( s ) ) ⁢ d s .

By c ∈ W 1 , ∞ ⁢ ( 0 , T ) and v ∈ X , assume f ⁢ ( x , t ) = c ⁢ ( t ) ⁢ v ⁢ ( x , t ) . Using arguments similar to the ones stated above (3.12) and (3.14), we have

(3.16) ∥ L ⁢ Q c ⁢ v ⁢ ( t ) ∥ D ⁢ ( L γ ) = ∥ ∫ 0 t K ⁢ ( t − s ) ⁢ L 1 + γ ⁢ ( c ⁢ ( s ) ⁢ v ⁢ ( s ) ) ⁢ d s ∥ ≤ C ⁢ ∥ c ∥ ∞ ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ∥ v ⁢ ( s ) ∥ D ⁢ ( L 1 + γ ) ⁢ d s ( by ( 3.7 ) ) ≤ C ⁢ T α 1 ⁢ ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ( by Lemma 2.3 ) ,

where C > 0 depends on ∥ c ∥ ∞ , and combining

d d ⁢ t ⁢ Q c ⁢ v ⁢ ( t ) = K ⁢ ( t ) ⁢ c ⁢ ( 0 ) ⁢ v ⁢ ( 0 ) + ∫ 0 t K ⁢ ( s ) ⁢ ( c ⁢ ( t − s ) ⁢ v ⁢ ( t − s ) ) t ⁢ d s ⁢ = by ( 3.10 ) ⁢ ∫ 0 t K ⁢ ( t − s ) ⁢ ( c ⁢ ( s ) ⁢ v ⁢ ( s ) ) s ⁢ d s ,

we obtain

(3.17) ∥ d d ⁢ t ⁢ Q c ⁢ v ⁢ ( t ) ∥ D ⁢ ( L γ ) ≤ ∫ 0 t ∥ K ⁢ ( t − s ) ⁢ L γ ⁢ ( c ⁢ ( s ) ⁢ v ⁢ ( s ) ) s ∥ ⁢ d s ( by the generalized Minkowski inequality ) ≤ C ⁢ ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∥ v ⁢ ( s ) ∥ D ⁢ ( L γ ) + ∥ v s ⁢ ( s ) ∥ D ⁢ ( L γ ) ) ⁢ d s ( by ( 3.7 ) ) ≤ C ⁢ T α 1 ⁢ ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ( by Lemma 2.3 ) ,

where C > 0 depends on ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) . So we arrive at

∥ Q c ⁢ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ C ⁢ T α 1 ⁢ ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) , ∥ Q c ⁢ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ C ⁢ T α 1 ⁢ ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ,

and also, the estimate ∥ Q c ⁢ v ∥ X ≤ C ⁢ T α 1 ⁢ ∥ v ∥ X holds, which means Q c ⁢ v ∈ X .

Then we define G ⁢ ( v ) ⁢ ( t ) = Q c ⁢ v ⁢ ( t ) + Q ⁢ f ⁢ ( t ) , where Q ⁢ f ⁢ ( t ) := ∫ 0 t K ⁢ ( t − s ) ⁢ f ⁢ ( s ) ⁢ d s . By the above discussions, we know the operator 𝐺 maps the Banach space 𝑋 into itself. Thus we will look for a fixed point of the operator G : X → X in order to prove the unique existence of the solution to the integral equation (3.11). Therefore, we have to obtain the contraction property of the map 𝐺. By induction, we have

G m ⁢ ( v ) ⁢ ( t ) = Q c m ⁢ v ⁢ ( t ) + ∑ k = 0 m − 1 Q c k ⁢ Q ⁢ f ⁢ ( t ) .

Here we denote Q c 0 = I .

Repeating calculations similar to (3.16) and (3.17), we have

∥ Q c 2 ⁢ v ⁢ ( t ) ∥ D ⁢ ( L 1 + γ ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ∥ Q c ⁢ v ⁢ ( s ) ∥ D ⁢ ( L 1 + γ ) ⁢ d s ≤ C 2 ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∫ 0 s ( s − τ ) α 1 − 1 ⁢ ∥ v ⁢ ( τ ) ∥ D ⁢ ( L 1 + γ ) ⁢ d τ ) ⁢ d s = C 2 ⁢ ∫ 0 t ( ∫ τ t ( t − s ) α 1 − 1 ⁢ ( s − τ ) α 1 − 1 ⁢ d s ) ⁢ ∥ v ⁢ ( τ ) ∥ D ⁢ ( L 1 + γ ) ⁢ d τ = ( C ⁢ Γ ⁢ ( α 1 ) ) 2 Γ ⁢ ( 2 ⁢ α 1 ) ⁢ ∫ 0 t ( t − τ ) 2 ⁢ α 1 − 1 ⁢ ∥ v ⁢ ( τ ) ∥ D ⁢ ( L 1 + γ ) ⁢ d τ ,

∥ d d ⁢ t ⁢ ( Q c 2 ⁢ v ) ∥ D ⁢ ( L γ ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∥ Q c ⁢ v ⁢ ( s ) ∥ D ⁢ ( L γ ) + ∥ d d ⁢ s ⁢ Q c ⁢ v s ⁢ ( s ) ∥ D ⁢ ( L γ ) ) ⁢ d s ≤ C 2 ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∫ 0 s ( s − τ ) α 1 − 1 ⁢ ( ∥ v ⁢ ( τ ) ∥ D ⁢ ( L γ ) + ∥ v τ ⁢ ( τ ) ∥ D ⁢ ( L γ ) ) ⁢ d τ ) ⁢ d s = ( C ⁢ Γ ⁢ ( α 1 ) ) 2 Γ ⁢ ( 2 ⁢ α 1 ) ⁢ ∫ 0 t ( t − τ ) 2 ⁢ α 1 − 1 ⁢ ( ∥ v ⁢ ( τ ) ∥ D ⁢ ( L γ ) + ∥ v τ ⁢ ( τ ) ∥ D ⁢ ( L γ ) ) ⁢ d τ .

By induction, we have

∥ Q c m ⁢ v ⁢ ( t ) ∥ D ⁢ ( L 1 + γ ) ≤ ( C ⁢ Γ ⁢ ( α 1 ) ) m Γ ⁢ ( m ⁢ α 1 ) ⁢ ∫ 0 t ( t − τ ) m ⁢ α 1 − 1 ⁢ ∥ v ⁢ ( τ ) ∥ D ⁢ ( L 1 + γ ) ⁢ d τ , ∥ d d ⁢ t ⁢ Q c m ⁢ v ⁢ ( t ) ∥ D ⁢ ( L γ ) ≤ ( C ⁢ Γ ⁢ ( α 1 ) ) m Γ ⁢ ( m ⁢ α 1 ) ⁢ ∫ 0 t ( t − τ ) m ⁢ α 1 − 1 ⁢ ( ∥ v ⁢ ( τ ) ∥ D ⁢ ( L γ ) + ∥ v τ ⁢ ( τ ) ∥ D ⁢ ( L γ ) ) ⁢ d τ .

By Lemma 2.3, we have Q c m ⁢ v ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ∩ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) and the estimates

(3.18) ∥ Q c m ⁢ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ ρ m ⁢ ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ,

(3.19) ∥ Q c ⁢ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ ρ m ⁢ ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ,

and also

∥ Q c m ⁢ v ∥ X ≤ ρ m ⁢ ∥ v ∥ X , v ∈ X , where ⁢ ρ m = ( C ⁢ Γ ⁢ ( α 1 ) ⁢ T α 1 ) m Γ ⁢ ( m ⁢ α 1 + 1 ) .

Therefore, for v 1 , v 2 ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ∩ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) , we obtain

∥ G m ⁢ ( v 1 ) − G m ⁢ ( v 2 ) ∥ X = ∥ Q c m ⁢ ( v 1 − v 2 ) ∥ X ≤ ρ m ⁢ ∥ v 1 − v 2 ∥ X .

It is easy to verify ρ m → 0 as m → ∞ . Therefore, we have | ρ m | < 1 for large m ∈ N . Therefore, the operator G m is a contraction mapping from 𝑋 into itself. Hence the mapping G m has a unique fixed point still denoted by v ∈ X , that is, G m ⁢ ( v ) = v , so we know G m + 1 ⁢ ( v ) = G ⁢ ( v ) . Since G m ⁢ ( G ⁢ ( v ) ) = G m + 1 ⁢ ( v ) = G ⁢ ( v ) , the point G ⁢ ( v ) is also a fixed point of the mapping G m . By the uniqueness of the fixed point of G m , we have Q c ⁢ v + Q ⁢ f = G ⁢ ( v ) = v , that is, the equation v = Q c ⁢ v + Q ⁢ f has a unique solution 𝑣 in C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ∩ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) . Moreover, we have

(3.20) v = G ⁢ ( v ) = G m ⁢ ( v ) = Q c m ⁢ v + ∑ k = 0 m − 1 Q c k ⁢ Q ⁢ f .

From the above discussion, we know Q ⁢ f ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ∩ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) . By (3.13) and (3.18), we have

∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ ∥ Q c m ⁢ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) + ∑ k = 0 m − 1 ∥ Q c k ⁢ Q ⁢ f ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ ρ m ⁢ ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) + ∑ k = 0 m − 1 ρ k ⁢ ∥ Q ⁢ f ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ ρ m ⁢ ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) + C ⁢ ∑ k = 0 m − 1 ρ k ⁢ T α 1 ⁢ ( 1 − γ ) ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L ) ) .

Taking sufficiently large m ∈ N such that ρ m < 1 , we have

(3.21) ∥ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) ≤ C ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L ) )

with C > 0 depending on 𝑇, Ω, α 1 , 𝛾 and ∥ c ∥ L ∞ ⁢ ( 0 , T ) .

As v ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) , we know that L ⁢ v ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L γ ) ) . Therefore, we have L ⁢ v ∈ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) with 0 ≤ γ < 1 from (2.1), and

(3.22) ∥ L ⁢ v ∥ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C ⁢ ∥ L ⁢ v ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L γ ) ) ≤ C ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) )

from (2.2) and (3.21). By the original equation ∑ j = 1 m q j ⁢ ∂ t α j v = − L ⁢ v + c ⁢ v + f , combining (3.4), (3.21) and (3.22), we see that ∑ j = 1 m q j ⁢ ∂ t α j v ∈ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) with the estimate

∥ ∑ j = 1 m q j ⁢ ∂ t α j v ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) ) + ∥ c ⁢ v ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) + ∥ f ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C ⁢ ∥ f ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) ) .

Therefore, we arrive at (3.5) from the above estimate and (3.22).

Finally, we prove estimate (3.6). From (3.20), (3.15) and (3.19), we know

∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ ∥ Q c m ⁢ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) + ∑ k = 0 m − 1 ∥ Q c k ⁢ Q ⁢ f ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ ρ m ⁢ ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) + ∑ k = 0 m − 1 ρ k ⁢ ∥ Q ⁢ f ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ ρ m ⁢ ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) + C ⁢ ∑ k = 0 m − 1 ρ k ⁢ T α 1 ⁢ ( 1 − γ ) ⁢ ∥ f ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) .

Taking sufficiently large m ∈ N such that ρ m < 1 , we have

∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ C ⁢ ∥ f ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) )

with C > 0 depending on 𝑇, Ω, α 1 , 𝛾 and ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) . Thus we complete the proof. ∎

Proof of Theorem 3.1

We split the solution 𝑢 of (1.1) into u = U + u 0 , where 𝑈 solves

(3.23) { ∑ j = 1 m q j ⁢ ∂ t α j U ⁢ ( x , t ) + L ⁢ U ⁢ ( x , t ) − c ⁢ ( t ) ⁢ U ⁢ ( x , t ) = F 0 ⁢ ( x , t ) , x ∈ Ω , 0 < t ≤ T , B σ ⁢ U ⁢ ( x , t ) = 0 , x ∈ ∂ Ω , 0 < t ≤ T , U ⁢ ( x , 0 ) = 0 , x ∈ Ω .

with F 0 ⁢ ( x , t ) = r ⁢ ( x ) ⁢ p ⁢ ( t ) − L ⁢ u 0 ⁢ ( x ) + c ⁢ ( t ) ⁢ u 0 ⁢ ( x ) . By (1.2)–(1.5), we have F 0 ∈ L ∞ ⁢ ( 0 , T ; D ⁢ ( L ) ) and the estimate

∥ F 0 ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) ) ≤ C ⁢ ( ∥ p ∥ ∞ ⁢ ∥ r ∥ H 2 ⁢ ( Ω ) + ∥ u 0 ∥ H 4 ⁢ ( Ω ) ) ,

where C > 0 depends on ∥ c ∥ L ∞ ⁢ ( 0 , T ) . By Lemma 3.2, IBVP (3.23) has a unique solution U ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) ) satisfying

L ⁢ U ∈ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) and ∑ j = 1 m q j ⁢ ∂ t α j U ∈ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) )

for 0 ≤ γ < 1 . Moreover, we have

∥ L ⁢ U ∥ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) + ∥ ∑ j = 1 m q j ⁢ ∂ t α j U ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ≤ C 1 ⁢ ∥ F 0 ∥ L ∞ ⁢ ( 0 , T ; H 2 ⁢ ( Ω ) ) ≤ C ⁢ ( ∥ p ∥ ∞ ⁢ ∥ r ∥ H 2 ⁢ ( Ω ) + ∥ u 0 ∥ H 4 ⁢ ( Ω ) )

with C > 0 depending on 𝑇, Ω, α 1 , 𝛾 and ∥ c ∥ L ∞ ⁢ ( 0 , T ) . Therefore, IBVP (1.1) admits a unique solution

u = U + u 0 ∈ C ⁢ ( [ 0 , T ] ; D ⁢ ( L 1 + γ ) )

satisfying

L ⁢ u ∈ C ⁢ ( [ 0 , T ] ; H 2 ⁢ γ ⁢ ( Ω ) ) and ∑ j = 1 m q j ⁢ ∂ t α j u ∈ L ∞ ⁢ ( 0 , T ; H 2 ⁢ γ ⁢ ( Ω ) ) ,

and estimate (3.1) holds.

Moreover, for f ∈ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ∩ C 0 ⁢ ( [ 0 , T ] ; L 2 ⁢ ( Ω ) ) and c ∈ W 1 , ∞ ⁢ ( 0 , T ) , we can obtain by Lemma 3.2 that

∥ ∂ t U ∥ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) ≤ C 2 ⁢ ∥ F 0 ∥ W 1 , ∞ ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) ≤ C ⁢ ( ∥ p ∥ W 1 , ∞ ⁢ ( 0 , T ) ⁢ ∥ r ∥ + ∥ u 0 ∥ H 2 ⁢ ( Ω ) ) ,

where C > 0 depends on α 1 , Ω, 𝑇, 𝛾 and ∥ c ∥ W 1 , ∞ ⁢ ( 0 , T ) . So estimate (3.2) also holds. Thus we complete the proof. ∎

4 Stability of the Inverse Problem

In this section, we give a conditional stability result for simultaneous inversion of two time-dependent coefficients with the help of the well-posedness result of the forward problem. To this end, we prepare the following lemmas with the Gronwall type inequalities.

Lemma 4.1

Let C , α > 0 and u , d ∈ L 1 ⁢ ( 0 , T ) be nonnegative functions satisfying

u ⁢ ( t ) ≤ C ⁢ d ⁢ ( t ) + C ⁢ ∫ 0 t ( t − s ) α − 1 ⁢ u ⁢ ( s ) ⁢ d s , t ∈ ( 0 , T ) .

Then we have

u ⁢ ( t ) ≤ C ⁢ d ⁢ ( t ) + C ⁢ ∫ 0 t ( t − s ) α − 1 ⁢ d ⁢ ( s ) ⁢ d s , t ∈ ( 0 , T ) .

Lemma 4.2

Let a , b , α > 0 and u ∈ L 1 ⁢ ( 0 , T ) be nonnegative functions satisfying

u ⁢ ( t ) ≤ a + b ⁢ ∫ 0 t ( t − s ) α − 1 ⁢ u ⁢ ( s ) ⁢ d s , a.e. ⁢ t ∈ ( 0 , T ) .

Then we have

u ⁢ ( t ) ≤ a ⁢ E α , 1 ⁢ ( ( b ⁢ Γ ⁢ ( α ) ) 1 / α ⁢ t α ) , a.e. ⁢ t ∈ ( 0 , T ) .

For the proofs of the above two lemmas, please refer to [12, Lemma 7.1.1 and Lemma 7.1.2 on pp. 188–189].

The main result in this section is the following stability result for the inverse potential and source coefficients problem.

Theorem 4.3

Assume conditions (1.2)–(1.5) hold. Let u i be the solution of (1.1) with c = c i and p = p i satisfying ∥ c i ∥ ∞ , ∥ p i ∥ ∞ ≤ M for i = 1 , 2 . Assume that there exists ν > 0 such that

(4.1) 0 < ν ≤ u 2 ( x 1 , t ) r ( x 2 ) − u 2 ( x 2 , t ) r ( x 1 ) = : m ( t ) , t ∈ ( 0 , T ) .

Then we have

∥ c 1 − c 2 ∥ ∞ + ∥ p 1 − p 2 ∥ ∞ ≤ C ⁢ ∑ i = 1 2 ∥ ∑ j = 1 m q j ⁢ ∂ t α j ( u 1 ⁢ ( x i , ⋅ ) − u 2 ⁢ ( x i , ⋅ ) ) ∥ ∞

and

(4.2) ∥ u 1 ⁢ ( x , t ) − u 2 ⁢ ( x , t ) ∥ L 2 ⁢ ( 0 , T ; D ⁢ ( L ) ) ≤ C ⁢ ( ∥ c 1 − c 2 ∥ L 2 ⁢ ( 0 , T ) + ∥ p 1 − p 2 ∥ L 2 ⁢ ( 0 , T ) ) .

Moreover, if

c ∈ V 1 := { v ∈ W 1 , ∞ ⁢ ( 0 , T ) , ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ) ≤ E 1 } , p ∈ V 2 := { v ∈ W 1 , ∞ ⁢ ( 0 , T ) ∩ C 0 ⁢ [ 0 , T ] , ∥ v ∥ W 1 , ∞ ⁢ ( 0 , T ) ≤ E 2 } ,

then we can obtain a nice conditional stability estimate as follows:

(4.3) ∥ c 1 − c 2 ∥ ∞ + ∥ p 1 − p 2 ∥ ∞ ≤ C ⁢ ∑ i = 1 2 ∑ j = 1 m ∥ u 1 ⁢ ( x i , ⋅ ) − u 2 ⁢ ( x i , ⋅ ) ∥ C ⁢ [ 0 , T ] 1 − α j ,

where C > 0 depending on E 1 , E 2 , 𝑀, α 1 , Ω, ∥ r ∥ H 2 ⁢ ( Ω ) and ∥ u 0 ∥ H 4 ⁢ ( Ω ) is a constant.

Proof

Let u i be the solution to (1.1) corresponding to c = c i and p = p i ( i = 1 , 2 ). We set u = u 1 − u 2 , c = c 1 − c 2 and p = p 1 − p 2 . Then 𝑢 solves

(4.4) { ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x , t ) + L ⁢ u ⁢ ( x , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x , t ) = c ⁢ ( t ) ⁢ u 2 ⁢ ( x , t ) + r ⁢ ( x ) ⁢ p ⁢ ( t ) , ( x , t ) ∈ Q T , B σ ⁢ u ⁢ ( x , t ) = 0 , ( x , t ) ∈ ∂ l Q T , u ⁢ ( x , 0 ) = 0 , x ∈ Ω .

Similar to (3.11), the solution of IBVP (4.4) satisfies the following integral equation:

u ⁢ ( ⋅ , t ) = ∫ 0 t K ⁢ ( t − s ) ⁢ ( c 1 ⁢ ( s ) ⁢ u ⁢ ( ⋅ , s ) ) ⁢ d s + ∫ 0 t K ⁢ ( t − s ) ⁢ ( c ⁢ ( s ) ⁢ u 2 ⁢ ( ⋅ , s ) + r ⁢ ( ⋅ ) ⁢ p ⁢ ( s ) ) ⁢ d s .

First we estimate ∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) . By (3.7), we obtain

(4.5) ∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) ≤ ∥ ∫ 0 t K ⁢ ( t − s ) ⁢ L ⁢ ( c 1 ⁢ ( s ) ⁢ u ⁢ ( ⋅ , s ) ) ⁢ d s ∥ + ∥ ∫ 0 t K ⁢ ( t − s ) ⁢ L ⁢ ( c ⁢ ( s ) ⁢ u 2 ⁢ ( ⋅ , s ) + r ⁢ ( ⋅ ) ⁢ p ⁢ ( s ) ) ⁢ d s ∥ ≤ C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ∥ u ⁢ ( ⋅ , s ) ∥ D ⁢ ( L ) ⁢ d s + C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∥ u 2 ⁢ ( ⋅ , s ) ∥ D ⁢ ( L ) ⁢ | c ⁢ ( s ) | + ∥ r ∥ D ⁢ ( L ) ⁢ | p ⁢ ( s ) | ) ⁢ d s

with C > 0 depending on α 1 , Ω and ∥ c 1 ∥ ∞ . In addition, we know from (3.1) that

(4.6) ∥ u 2 ∥ C ⁢ ( [ 0 , T ] ; D ⁢ ( L ) ) ≤ C ⁢ ( ∥ p 2 ∥ ∞ ⁢ ∥ r ∥ H 2 ⁢ ( Ω ) + ∥ u 0 ∥ H 4 ⁢ ( Ω ) )

with C > 0 depending on α 1 , Ω , T , ∥ c 2 ∥ ∞ . Combing (4.6), (1.4), (1.5) and ∥ p i ∥ ∞ ≤ M , we can rewrite (4.5) as follows:

∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ∥ u ⁢ ( ⋅ , s ) ∥ D ⁢ ( L ) ⁢ d s + C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s .

Denote d ⁢ ( t ) = ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s ; then, by Lemma 4.1, we obtain

∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) ≤ C ⁢ d ⁢ ( t ) + ∫ 0 t ( t − s ) α 1 − 1 ⁢ d ⁢ ( s ) ⁢ d s , t ∈ ( 0 , T ) .

Since

∫ 0 t ( t − s ) α 1 − 1 ⁢ d ⁢ ( s ) ⁢ d s = ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( ∫ 0 s ( s − τ ) α 1 − 1 ⁢ ( | c ⁢ ( τ ) | + | p ⁢ ( τ ) | ) ⁢ d τ ) ⁢ d s = ∫ 0 t ( ∫ τ t ( t − s ) α 1 − 1 ⁢ ( s − τ ) α 1 − 1 ⁢ d s ) ⁢ ( | c ⁢ ( τ ) | + | p ⁢ ( τ ) | ) ⁢ d τ = B ⁢ ( α 1 , α 1 ) ⁢ ∫ 0 t ( t − τ ) 2 ⁢ α 1 − 1 ⁢ ( | c ⁢ ( τ ) | + | p ⁢ ( τ ) | ) ⁢ d τ ≤ T α 1 ⁢ B ⁢ ( α 1 , α 1 ) ⁢ ∫ 0 t ( t − τ ) α 1 − 1 ⁢ ( | c ⁢ ( τ ) | + | p ⁢ ( τ ) | ) ⁢ d τ ≤ C ⁢ d ⁢ ( t ) ,

we obtain ∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) ≤ C ⁢ d ⁢ ( t ) , t ∈ ( 0 , T ) , that is,

∥ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s , t ∈ ( 0 , T ) .

By the Young inequality for the convolution, we have

∫ 0 T ∥ u ⁢ ( t ) ∥ D ⁢ ( L ) 2 ⁢ d t ≤ C ⁢ ∫ 0 T ( ∫ 0 t ( t − s ) α 1 − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s ) 2 ⁢ d t ≤ C ⁢ T 2 ⁢ α 1 ⁢ ( ∥ c ∥ L 2 ⁢ ( 0 , T ) + ∥ p ∥ L 2 ⁢ ( 0 , T ) ) 2 .

That means (4.2) is true. In the same way, we have that, for 0 ≤ γ < 1 ,

(4.7) ∥ L ⁢ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L γ ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s , t ∈ ( 0 , T ) .

Let d / 4 < γ < 1 . By the Sobolev embedding, combining (4.7), we have

(4.8) | L ⁢ u ⁢ ( x i , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x i , t ) | ≤ C ⁢ ∥ L ⁢ u ⁢ ( ⋅ , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( ⋅ , t ) ∥ H 2 ⁢ γ ⁢ ( Ω ) ≤ C ⁢ ∥ L ⁢ u ⁢ ( ⋅ , t ) ∥ H 2 ⁢ γ ⁢ ( Ω ) + C ⁢ ∥ u ⁢ ( ⋅ , t ) ∥ H 2 ⁢ γ ⁢ ( Ω ) ≤ C ⁢ ∥ L ⁢ u ⁢ ( ⋅ , t ) ∥ D ⁢ ( L γ ) ≤ C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s , t ∈ ( 0 , T ) , i = 1 , 2 ,

where we use the Sobolev embedding H 2 ⁢ γ ⁢ ( Ω ) ↪ C ⁢ [ Ω ̄ ] for γ > d / 4 in the first inequality and the fact

∥ u ∥ H 2 ⁢ γ ⁢ ( Ω ) ≲ ∥ u ∥ D ⁢ ( L γ ) ≲ ∥ L ⁢ u ∥ D ⁢ ( L γ )

in the last inequality. Then, replacing x = x i , i = 1 , 2 , in the first equation of (4.4), i.e.

{ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x 1 , t ) + L ⁢ u ⁢ ( x 1 , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x 1 , t ) = u 2 ⁢ ( x 1 , t ) ⁢ c ⁢ ( t ) + r ⁢ ( x 1 ) ⁢ p ⁢ ( t ) , 0 < t ≤ T , ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x 2 , t ) + L ⁢ u ⁢ ( x 2 , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x 2 , t ) = u 2 ⁢ ( x 2 , t ) ⁢ c ⁢ ( t ) + r ⁢ ( x 2 ) ⁢ p ⁢ ( t ) , 0 < t ≤ T ,

we obtain

(4.9) c ⁢ ( t ) = 1 m ⁢ ( t ) ⁢ [ P ⁢ [ u ] ⁢ ( x 1 , t ) ⁢ r ⁢ ( x 2 ) − P ⁢ [ u ] ⁢ ( x 2 , t ) ⁢ r ⁢ ( x 1 ) ] ,

p ⁢ ( t ) = 1 m ⁢ ( t ) ⁢ [ P ⁢ [ u ] ⁢ ( x 2 , t ) ⁢ u 2 ⁢ ( x 1 , t ) − P ⁢ [ u ] ⁢ ( x 1 , t ) ⁢ u 2 ⁢ ( x 2 , t ) ] ,

where

(4.10) { P ⁢ [ u ] ⁢ ( x 1 , t ) = ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x 1 , t ) + L ⁢ u ⁢ ( x 1 , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x 1 , t ) , P ⁢ [ u ] ⁢ ( x 2 , t ) = ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x 2 , t ) + L ⁢ u ⁢ ( x 2 , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x 2 , t ) .

From condition (4.1), we obtain

| m ⁢ ( t ) | = | u 2 ⁢ ( x 1 , t ) ⁢ r ⁢ ( x 2 ) − u 2 ⁢ ( x 2 , t ) ⁢ r ⁢ ( x 1 ) | > ν > 0 .

Therefore, combining (4.8), (4.9) and (4.10), we arrive at

| c ⁢ ( t ) | ≤ C ⁢ ∑ i = 1 2 | ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x i , t ) | + C ⁢ ∑ i = 1 2 | L ⁢ u ⁢ ( x i , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x i , t ) | ≤ C ⁢ ∑ i = 1 2 ∥ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x i , t ) ∥ L ∞ ⁢ ( 0 , T ) + C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s , t ∈ ( 0 , T ) .

Using the same procedure as in the above discussion, we can obtain that

| p ⁢ ( t ) | ≤ C ⁢ ∑ i = 1 2 ∥ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x i , t ) ∥ L ∞ ⁢ ( 0 , T ) + C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s .

So we have

| c ⁢ ( t ) | + | p ⁢ ( t ) | ≤ C ⁢ ∑ i = 1 2 ∥ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x i , t ) ∥ L ∞ ⁢ ( 0 , T ) + C ⁢ ∫ 0 t ( t − s ) α 1 ⁢ ( 1 − γ ) − 1 ⁢ ( | c ⁢ ( s ) | + | p ⁢ ( s ) | ) ⁢ d s .

Applying Lemma 4.2, we have

| c ⁢ ( t ) | + | p ⁢ ( t ) | ≤ C ⁢ ∑ i = 1 2 ∥ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x i , t ) ∥ L ∞ ⁢ ( 0 , T ) .

Finally, we prove the conditional stability estimate (4.3). By Theorem 3.1, we know ∂ t u ∈ L ∞ ⁢ ( 0 , T ; D ⁢ ( L γ ) ) with 0 ≤ γ < 1 . Combining the Sobolev embedding, we have ∂ t u ⁢ ( x i , ⋅ ) ∈ L ∞ ⁢ ( 0 , T ) for γ > d / 4 . Therefore, estimate (4.3) is straightforward by Lemma 2.4. Thus we complete the proof. ∎

Remark 1

In this paper, we use the additional condition 0 < ν ≤ u 2 ( x 1 , t ) r ( x 2 ) − u 2 ( x 2 , t ) r ( x 1 ) = : m ( t ) to identify simultaneously the time coefficients c ⁢ ( t ) and p ⁢ ( t ) ; although we just know the measured data of u ⁢ ( x i , t ) , r ⁢ ( x i ) , i = 1 , 2 , it is rational to suppose m ⁢ ( t ) be known roughly; thus the condition | m ⁢ ( t ) | ≥ ν > 0 can be checked approximately.

5 Ill-Posedness and Regularization Method

In this section, we will account for the ill-posedness of the inverse problem considered in the paper. Then we introduce a non-stationary iterative regularization method to overcome its ill-posedness.

Based on Theorem 3.1, we firstly define a forward operator

(5.1) F : ( c ⁢ ( t ) , p ⁢ ( t ) ) ∈ D ⁢ ( F ) ↦ ( u ⁢ ( x 1 , t ) , u ⁢ ( x 2 , t ) ) ∈ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) ,

where D ⁢ ( F ) = H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) , u ⁢ ( x , t ) is the solution of (1.1) and for the choice of x i , one can refer to the description of (IP1) and (IP2). Thus the inverse problem is formulated into solving the following abstract operator equation:

F ( c , p ) = ( u ( x 1 , t ) , u ( x 2 , t ) ) = : ( h 1 ( t ) , h 2 ( t ) ) = h ( t ) .

Let u ⁢ ( x , t ) = v ⁢ ( x , t ) + u 0 ⁢ ( x ) . Then v ⁢ ( x , t ) solves problem

{ ∑ j = 1 m q j ⁢ ∂ t α j v ⁢ ( x , t ) + L ⁢ v ⁢ ( x , t ) − c ⁢ ( t ) ⁢ v ⁢ ( x , t ) = f ⁢ ( x , t ) , x ∈ Ω , 0 < t ≤ T , B σ ⁢ v ⁢ ( x , t ) = 0 , x ∈ ∂ Ω , 0 < t ≤ T , v ⁢ ( x , 0 ) = 0 , x ∈ Ω .

where f ⁢ ( x , t ) = r ⁢ ( x ) ⁢ p ⁢ ( t ) − L ⁢ u 0 ⁢ ( x ) + c ⁢ ( t ) ⁢ u 0 ⁢ ( x ) . By Lemma 3.2, for γ > d / 4 , we know that ∂ t α 1 v ⁢ ( x i , t ) ∈ L ∞ ⁢ ( 0 , T ) , i = 1 , 2 , by the Sobolev embedding. Noting v ⁢ ( x , 0 ) = 0 , we know from the book [20] that there exists a function ξ ⁢ ( t ) ∈ L ∞ ⁢ ( 0 , T ) such that v ⁢ ( x i , t ) = I 0 + α 1 ⁢ ξ ⁢ ( t ) . From [8, Theorem 2.1], we have v ⁢ ( x i , t ) ∈ H α 1 ⁢ ( 0 , T ) and further

u ⁢ ( x i , t ) = v ⁢ ( x i , t ) + u 0 ⁢ ( x i ) ∈ H α 1 ⁢ ( 0 , T ) .

As H α 1 ⁢ ( 0 , T ) ↪ L 2 ⁢ ( 0 , T ) compactly, the operator F : D ⁢ ( F ) ↦ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) is compact.

Theorem 5.1

Under the conditions of Theorem 4.3, the operator ℱ is continuous from D ⁢ ( F ) into L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) .

Proof

Let u i be the solution to (1.1) corresponding to c = c i and p = p i ( i = 1 , 2 ). We set u = u 1 − u 2 , c = c 1 − c 2 and p = p 1 − p 2 . Then 𝑢 solves

{ ∑ j = 1 m q j ⁢ ∂ t α j u ⁢ ( x , t ) + L ⁢ u ⁢ ( x , t ) − c 1 ⁢ ( t ) ⁢ u ⁢ ( x , t ) = c ⁢ ( t ) ⁢ u 2 ⁢ ( x , t ) + r ⁢ ( x ) ⁢ p ⁢ ( t ) , ( x , t ) ∈ Q T , B σ ⁢ u ⁢ ( x , t ) = 0 , ( x , t ) ∈ ∂ l Q T , u ⁢ ( x , 0 ) = 0 , x ∈ Ω .

By Theorem 4.3, we have the following estimate:

∥ u 1 ⁢ ( x , t ) − u 2 ⁢ ( x , t ) ∥ L 2 ⁢ ( 0 , T ; D ⁢ ( L ) ) ≤ C ⁢ ( ∥ c 1 − c 2 ∥ L 2 ⁢ ( 0 , T ) + ∥ p 1 − p 2 ∥ L 2 ⁢ ( 0 , T ) ) .

So we immediately obtain the continuity of ℱ from D ⁢ ( F ) ↦ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) . ∎

Therefore, by the continuity and compactness of the operator ℱ and the closed convexity of D ⁢ ( F ) , we know that the inverse problem considered in the paper is ill-posed based on the authoritative book [6].

Now we deduce a numerical algorithm for simultaneously recovering the time-dependent potential function c ⁢ ( t ) and the time-dependent source term p ⁢ ( t ) of IBVP (1.1) by the additional measured data u ⁢ ( x i , t ) , 0 < t < T , i = 1 , 2 . We firstly transform the inverse problem into a variational problem. Then the Levenberg–Marquardt method is introduced, and we use it to minimize the variational problem.

Let ( c * , p * ) ∈ H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) be a suitable guess of ( c , p ) . In order to ensure a stable numerical reconstruction of ( c ⁢ ( t ) , p ⁢ ( t ) ) , we introduce the following variational problem with a high-order Tikhonov regularization term:

(5.2) min ( c , p ) ∈ D ⁢ ( F ) ⁡ J ̃ ⁢ ( c , p ) = 1 2 ⁢ ∥ u ⁢ ( x 1 , t ; c , p ) − h 1 δ ∥ L 2 ⁢ ( 0 , T ) 2 + 1 2 ⁢ ∥ u ⁢ ( x 2 , t ; c , p ) − h 2 δ ∥ L 2 ⁢ ( 0 , T ) 2 + μ 2 ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν 2 ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2

where μ , ν > 0 are regularization parameters, and h i δ , i = 1 , 2 , are noisy functions of h i .

Lemma 5.2

Lemma 5.2 ([4])

Assume that 𝐸 is a uniformly convex Banach space. Let { x n } be a sequence in 𝐸 such that x n ⇀ x weakly in σ ⁢ ( E , E ′ ) and lim sup ∥ x n ∥ ≤ ∥ x ∥ . Then x n → x strongly.

Proposition 5.3

Under the conditions of Theorem 4.3, there exists at least one minimizer ( c μ δ , p ν δ ) ∈ D ⁢ ( F ) for the variational problem (5.2).

Proof

Since the functional J ̃ is nonnegative, there exists d = inf ( c , p ) ∈ D ⁢ ( F ) J ̃ ⁢ ( c , p ) . Thus there exists a sequence ( c n , p n ) ∈ H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) such that J ̃ ⁢ ( c n , p n ) → d as n → ∞ . Therefore, we obtain that μ ⁢ ∥ c n − c * ∥ H 1 ⁢ ( 0 , T ) 2 and ν ⁢ ∥ p n − p * ∥ H 1 ⁢ ( 0 , T ) 2 are bounded. That shows that { c n } and { p n } are bounded in H 1 ⁢ ( 0 , T ) ; then there exists a subsequence, still denoted by c n , p n , such that ( c n , p n ) ⇀ ( c μ δ , p ν δ ) in H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) and ( c n , p n ) → ( c μ δ , p ν δ ) in L 2 ⁢ ( 0 , T ) . Based on (4.2) of Theorem 4.3 and the Sobolev embedding, we have u ⁢ ( x i , t ; c n , p n ) → u ⁢ ( x i , t ; c μ δ , p ν δ ) in L 2 ⁢ ( 0 , T ) , i.e.

∥ u ⁢ ( x i , t ; c n , p n ) − h i δ ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) 2 → ∥ u ⁢ ( x i , t ; c μ δ , p ν δ ) − h i δ ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) 2 , n → ∞ , i = 1 , 2 .

Based on the weak lower semicontinuity of the H 1 -norm, we have

μ ⁢ ∥ c μ δ − c * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ lim inf n → ∞ μ ⁢ ∥ c n − c * ∥ H 1 ⁢ ( 0 , 1 ) 2 , ν ⁢ ∥ p ν δ − p * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ lim inf n → ∞ ν ⁢ ∥ p n − p * ∥ H 1 ⁢ ( 0 , 1 ) 2 .

Therefore, we have

d ≤ J ̃ ⁢ ( c μ δ , p ν δ ) ≤ lim inf n → ∞ J ̃ ⁢ ( c n , p n ) = d .

Then ( c μ δ , p ν δ ) is a minimizer. ∎

Proposition 5.4

Under the conditions of Theorem 4.3, assume F ⁢ ( c , p ) = ( h 1 , h 2 ) and the noisy data h i δ ∈ L 2 ⁢ ( 0 , T ) satisfying ∥ h i δ − h i ∥ L 2 ⁢ ( 0 , T ) ≤ δ for i = 1 , 2 , and let μ ⁢ ( δ ) , ν ⁢ ( δ ) satisfy | μ ⁢ ( δ ) / ν ⁢ ( δ ) | ≤ C , where 𝐶 is a positive constant, and μ ⁢ ( δ ) , ν ⁢ ( δ ) → 0 , δ 2 / μ ⁢ ( δ ) , δ 2 / ν ⁢ ( δ ) → 0 as δ → 0 . Then the minimizer c μ δ , p ν δ of variational problem (5.2) is convergent, i.e. ( c μ ⁢ ( δ ) δ , p ν ⁢ ( δ ) δ ) → ( c , p ) in H 1 ⁢ ( 0 , T ) as δ → 0 .

Proof

Let δ k be any sequence such that δ k → 0 , and denote μ k = μ ⁢ ( δ k ) , ν k = ν ⁢ ( δ k ) . By the definition of the minimizer c μ δ , p ν δ , we have

(5.3) ∥ F ⁢ ( c μ k δ k , p ν k δ k ) − h δ k ∥ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) 2 + μ k ⁢ ∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν k ⁢ ∥ p μ k δ k − p * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ ∥ F ⁢ ( c , p ) − h δ k ∥ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) 2 + μ k ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν k ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ δ k 2 + μ k ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν k ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2 .

So we know

∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ δ k 2 / μ k + ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν k / μ k ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2 , ∥ p ν k δ k − p * ∥ H 1 ⁢ ( 0 , T ) 2 ≤ δ k 2 / ν k + ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2 + μ k / ν k ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 .

Also we obtain

lim sup δ k → 0 ∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) ≤ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) + C ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) , lim sup δ k → 0 ∥ p ν k δ k − p * ∥ H 1 ⁢ ( 0 , T ) ≤ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) + C ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T )

by the conditions for 𝜇, 𝜈. This illustrates that ∥ c μ k δ k ∥ H 1 ⁢ ( 0 , T ) and ∥ p ν k δ k ∥ H 1 ⁢ ( 0 , T ) are bounded and have weak convergent subsequences by the reflexivity of H 1 ⁢ ( 0 , T ) and also denotes ( c μ k δ k , p ν k δ k ) such that

( c μ k δ k , p ν k δ k ) ⇀ ( z 1 , z 2 ) in ⁢ H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) .

Since H 1 ⁢ ( 0 , T ) ↪ L 2 ⁢ ( 0 , T ) compactly, there exists the subsequence, still denoted by ( c μ k δ k , p ν k δ k ) , such that

( c μ k δ k , p ν k δ k ) → ( z 1 , z 2 ) in L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) .

By estimate (4.2), we have

F ⁢ ( c μ k δ k , p ν k δ k ) → F ⁢ ( z 1 , z 2 ) in L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) .

From (5.3), we have

∥ F ⁢ ( c μ k δ k , p ν k δ k ) − h δ k ∥ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) 2 ≤ δ k 2 + μ k ⁢ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) 2 + ν k ⁢ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) 2 ,

and thus

lim δ k → 0 F ⁢ ( c μ k δ k , p ν k δ k ) = h and F ⁢ ( z 1 , z 2 ) = h .

By the uniqueness of F ⁢ ( c , p ) = h , we know that ( z 1 , z 2 ) = ( c , p ) . According to the weak lower semicontinuity of the norm in Hilbert space, we have

∥ c − c * ∥ H 1 ⁢ ( 0 , T ) ≤ lim inf δ k → 0 ∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) ≤ lim sup δ k → 0 ∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) ≤ ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) , ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) ≤ lim inf δ k → 0 ∥ p ν k δ k − p * ∥ H 1 ⁢ ( 0 , T ) ≤ lim sup δ k → 0 ∥ p ν k δ k − p * ∥ H 1 ⁢ ( 0 , T ) ≤ ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) ,

and hence we know that

lim δ k → 0 ∥ c μ k δ k − c * ∥ H 1 ⁢ ( 0 , T ) = ∥ c − c * ∥ H 1 ⁢ ( 0 , T ) and lim δ k → 0 ∥ p ν k δ k − p * ∥ H 1 ⁢ ( 0 , T ) = ∥ p − p * ∥ H 1 ⁢ ( 0 , T ) .

Combing with the weak convergence of ( c μ k δ k , p ν k δ k ) in H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) , we obtain that ( c μ k δ k , p ν k δ k ) → ( c , p ) by Lemma 5.2. From the uniqueness of F ⁢ ( c , p ) = h , we arrive at ( c μ k δ k , p ν k δ k ) → ( c , p ) in H 1 ⁢ ( 0 , T ) × H 1 ⁢ ( 0 , T ) . ∎

In the following, we use the modified Levenberg–Marquardt method to minimize problem (5.2). The Levenberg–Marquardt method was first introduced by [22, 32], and it is a kind of the Newton type method. From physical considerations, we known that h δ is a reasonably close approximation of some ideal h = F ⁢ ( c , p ) in the range of ℱ. We assume that ( c * , p * ) is an approximation of ( c , p ) ; then the nonlinear mapping F ⁢ ( c , p ) in (5.1) can be replaced approximatively by its linearization around ( c * , p * ) , i.e.

F ⁢ ( c , p ) ≈ F ⁢ ( c * , p * ) + F c ⁢ ( c * , p * ) ⁢ ( c − c * ) + F p ⁢ ( c * , p * ) ⁢ ( p − p * ) .

Then the nonlinear inverse problem F ⁢ ( c , p ) = h δ can be transformed into a linear inverse problem

F c ⁢ ( c * , p * ) ⁢ ( c − c * ) + F p ⁢ ( c * , p * ) ⁢ ( p − p * ) = h δ − F ⁢ ( c * , p * ) .

Therefore, it is easily seen that the problem of minimizing (5.2) is approximated by minimizing

J ⁢ ( c , p ) = 1 2 ⁢ ∥ F c ⁢ ( c * , p * ) ⁢ δ ⁢ c + F p ⁢ ( c * , p * ) ⁢ δ ⁢ p − ( h δ − F ⁢ ( c * , p * ) ) ∥ L 2 ⁢ ( 0 , T ) × L 2 ⁢ ( 0 , T ) 2 + μ 2 ⁢ ∥ δ ⁢ c ∥ H 1 ⁢ ( 0 , T ) 2 + ν 2 ⁢ ∥ δ ⁢ p ∥ H 1 ⁢ ( 0 , T ) 2 ,

where δ ⁢ c = c − c * , δ ⁢ p = p − p * .

Therefore, simultaneously recovering the time-dependent potential and source term problem is transformed into solving a variational problem

(5.4) J ⁢ ( c μ δ , p ν δ ) = min ( c , p ) ∈ D ⁢ ( F ) ⁡ J ⁢ ( c , p ) .

In the following, we focus on the numerical method for solving problem (5.4). We can find many algorithms to solve it, such as the traditional gradient type methods, but we need to calculate the gradient of the minimizing problem by the information of its adjoint problem. However, we have to use a lot of effort to compute the Hessian operator. Therefore, we here use the finite-dimensional discrete method to approximate the infinite-dimensional variational problem in order to avoid great many calculations. To this end, suppose that { ϕ n ⁢ ( t ) , n = 1 , 2 , … , ∞ } is a set of basis functions in H 1 ⁢ ( 0 , T ) , let

c ⁢ ( t ) ≈ c N ⁢ ( t ) = ∑ n = 1 N a n ⁢ ϕ n ⁢ ( t ) and p ⁢ ( t ) ≈ p M ⁢ ( t ) = ∑ m = 1 M a m ′ ⁢ ϕ m ⁢ ( t ) ,

where N , M ∈ N and c N ⁢ ( t ) , p M ⁢ ( t ) are the 𝑁- and 𝑀-dimensional approximate solutions to c ⁢ ( t ) , p ⁢ ( t ) , respectively, and a n , n = 1 , 2 , … , N , and a m ′ , m = 1 , 2 , … , M , are the expansion coefficients. We set

Φ N = span ⁡ { ϕ 1 , ϕ 2 , … , ϕ N } , Φ M = span ⁡ { ϕ 1 , ϕ 2 , … , ϕ M }

and two vectors

a = ( a 1 , a 2 , … , a N ) ∈ R N , a ′ = ( a 1 ′ , a 2 ′ , … , a M ′ ) ∈ R M .

We identify an approximation c N ⁢ ( t ) ∈ Φ N with a vector a ∈ R N and p M ⁢ ( t ) ∈ Φ M with a ′ ∈ R M .

Here we modify the traditional Levenberg–Marquardt method that always solves the single-parameter inversion. We transform two-parameter inversion in finite-dimensional space into a one-parameter inversion problem; see also [45]. Based on the above discussions, denote b = ( a , a ′ ) ∈ R N + M ; by setting

( u ⁢ ( x 1 , t ; b ) , u ⁢ ( x 2 , t ; b ) ) = ( u ⁢ ( x 1 , t ; c N , p M ) , u ⁢ ( x 2 , t ; c N , p M ) ) = F ⁢ ( c N , p M )

as a unique solution of the forward problem, a feasible way for numerical solution is to solve the following minimization problem:

(5.5) min b ∈ R N + M { ∥ ∇ b T u ( x 1 , t ; b * ) δ b − ( h 1 δ − u ( x 1 , t ; b * ) ) ∥ L 2 ⁢ ( 0 , T ) 2 + ∥ ∇ b T u ( x 2 , t ; b * ) δ b − ( h 2 δ − u ( x 2 , t ; b * ) ) ∥ L 2 ⁢ ( 0 , T ) 2 + δ b D ( δ b ) T } ,

where D = diag ⁡ ( ( μ ⁢ ( ϕ i , ϕ j ) H 1 ⁢ ( Ω ) ) N × N , ( ν ⁢ ( ϕ i , ϕ j ) H 1 ⁢ ( Ω ) ) M × M ) , δ ⁢ b = b − b * and b T denotes the transpose of 𝒃.

Next, we give an iterative algorithm for determining an approximate coefficient 𝒃. For any given b j ∈ R N + M , we set

b j + 1 = b j + δ ⁢ b j , j = 1 , 2 , … ,

where δ ⁢ b j denotes a perturbation of b j for each 𝑗 and 𝑗 is the number of iterations.

We discretize the time domain [ 0 , T ] with 0 = t 0 < t 1 < ⋯ < t L = T ; then the L 2 -norm can be reduced to the discrete Euclid norm and the variational problem (5.5) at the 𝑗th step becomes

(5.6) min δ ⁢ b j ∈ R N + M ⁡ { T L ⁢ ( ∥ δ ⁢ b j ⁢ F 1 T − ( W 1 − U 1 ) ∥ 2 2 + ∥ δ ⁢ b j ⁢ F 2 T − ( W 2 − U 2 ) ∥ 2 2 ) + δ ⁢ b j ⁢ D ⁢ ( δ ⁢ b j ) T } ,

where

F l = ( f i ⁢ j l ) L × ( N + M ) , f i ⁢ j l = u ⁢ ( x l , t i ; ( b 1 k , … , b j k + τ , … , b N + M k ) ) − u ⁢ ( x l , t i ; b k ) τ , l = 1 , 2 ,

𝜏 denotes the numerical differential step, and

U l = ( u ⁢ ( x l , t 1 ; b k ) , u ⁢ ( x l , t 2 ; b k ) , … , u ⁢ ( x l , t L ; b k ) ) , l = 1 , 2 , W l = ( h l δ ⁢ ( t 1 ) , h l δ ⁢ ( t 2 ) , … , h l δ ⁢ ( t L ) ) , l = 1 , 2 .

By the variational theory, problem (5.6) is equivalent to the following normal equation:

(5.7) ( L T ⁢ D + F 1 T ⁢ F 1 + F 2 T ⁢ F 2 ) ⁢ δ ⁢ b j = F 1 T ⁢ ( W 1 T − U 1 T ) + F 2 T ⁢ ( W 2 T − U 2 T ) .

Thus the normal equation (5.7) can be solved by

δ ⁢ b j = ( M T ⁢ D + F 1 T ⁢ F 1 + F 2 T ⁢ F 2 ) − 1 ⁢ ( F 1 T ⁢ ( W 1 T − U 1 T ) + F 2 T ⁢ ( W 2 T − U 2 T ) ) .

6 Numerical Experiments

In this section, we present three numerical results of recovering the time potential term and time source term in problem (1.1) with σ = 1 in one- and two-dimensional cases to show the effectiveness of the Levenberg–Marquardt method.

The noisy data is generated by adding a random perturbation, i.e.

h i δ = h i + ϵ ⁢ h i ⋅ ( 2 ⋅ rand ⁡ ( size ⁡ ( h i ) ) − 1 ) , i = 1 , 2 .

The corresponding noise level is calculated by δ i = ∥ h i δ − h i ∥ L 2 ⁢ ( 0 , T ) , i = 1 , 2 . Also, we denote δ = δ 1 2 + δ 2 2 .

To show the accuracy of numerical solution, we compute the approximate L 2 error denoted by

re c k = ∥ c k ⁢ ( t ) − c ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) ∥ c ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) , re p k = ∥ p k ⁢ ( t ) − p ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) ∥ p ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) ,

where ( c k ⁢ ( t ) , p k ⁢ ( t ) ) are the coefficient terms reconstructed at the 𝑘th iteration, and ( c ⁢ ( t ) , p ⁢ ( t ) ) are the exact solutions. And let re k = 1 / 2 ⁢ ( re c k + re p k ) .

The residual E k at the 𝑘th iteration is given by

E k = ( ∥ u ⁢ ( x 1 , t ; c k , p k ) − h 1 δ ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) 2 + ∥ u ⁢ ( x 2 , t ; c k , p k ) − h 2 δ ⁢ ( t ) ∥ L 2 ⁢ ( 0 , T ) 2 ) 1 / 2 .

In an iteration algorithm, the most important work is to find a suitable stopping rule. In this study, we use the well-known discrepancy principle [36], i.e. we choose 𝑘 satisfying the inequality E k ≤ η ⁢ δ < E k − 1 , where η > 1 is a constant and can be taken heuristically to be 1.01, as suggested by Hanke and Hansen [10]. If the noise level is 0, then we take k = 20 for the following examples.

Without loss of generality, we set T = 1 , m = 3 and q j = 1 ( j = 1 , 2 , 3 ). The grid points on [ 0 , 1 ] and [ 0 , T ] are both 51 when solving the direct problem by finite difference method in [37] and also [45]. We consider the following three examples.

6.1 One-Dimensional Case

Without loss of generality, the space domain Ω is taken as ( 0 , 1 ) , and we fix x 1 = 0 , x 2 = 1 as observed data points. Let the elliptic operator be L = − Δ and the basis function spaces Φ N = span ⁡ { 2 ⁢ sin ⁡ ( π ⁢ t ) , … , 2 ⁢ sin ⁡ ( N ⁢ π ⁢ t ) } , and Φ M = span ⁡ { 2 ⁢ sin ⁡ ( π ⁢ t ) , … , 2 ⁢ sin ⁡ ( M ⁢ π ⁢ t ) } .

Example 1

Suppose the unknown time potential term is c ⁢ ( t ) = exp ⁡ ( − t ) ⁢ sin ⁡ ( 4 ⁢ π ⁢ t ) and unknown time source is p ⁢ ( t ) = 4 ⁢ t 3 ⁢ ( 1 − t ) . Taking the space source function r ⁢ ( x ) = cos ⁡ ( 2 ⁢ π ⁢ x ) and the initial state u 0 ⁢ ( x ) = cos ⁡ ( π ⁢ x ) , the additional measured data u ⁢ ( x 1 , t ) , u ⁢ ( x 2 , t ) are obtained by solving direct problem (1.1) by using the finite difference method.

The numerical results for Example 1 by using the discrepancy principle for various noise levels in the cases of α = ( 0.9 , 0.7 , 0.2 ) and α = ( 0.8 , 0.5 , 0.4 ) are shown in Figures 1–2. We choose the initial guess as

( c 0 , p 0 ) = ( c * , p * ) = ( 0 , 0 ) ,

the dimension N = 6 and M = 5 , the numerical differential step τ = 0.02 , and the regularization parameters chosen by μ = ν = 0.1 ⁢ δ 3 / 2 .

Figure 1

The numerical results for Example 1 for α = ( 0.9 , 0.7 , 0.2 ) .

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Figure 2

The numerical results for Example 1 for α = ( 0.8 , 0.5 , 0.4 ) .

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

In order to demonstrate the influence of different fractional orders on numerical results for Example 1, we choose α = ( 0.9 , 0.5 , 0.1 ) , ( 0.9 , 0.3 , 0.2 ) , ( 0.8 , 0.7 , 0.6 ) , ( 0.3 , 0.2 , 0.1 ) and list the relative errors of the reconstructed solution c k and p k for different noise levels in Table 1. With fixed 𝜖, we can observe that the fractional order has little effect on the accuracy of the numerical inversion results. With fixed 𝜶, we can see that the numerical error is decreasing as the level of noise becomes smaller. Also, we find that our algorithm has a good reconstruction effect on smooth unknown functions and the source term inversion is slightly better than the potential function reconstruction from Figures 1–2 and Table 1.

Table 1

The relative errors of Example 1 for different orders 𝜶 and noise levels 𝜖.

𝜖 \ 𝜶	Error	( 0.9 , 0.5 , 0.1 )	( 0.9 , 0.3 , 0.2 )	( 0.8 , 0.7 , 0.6 )	( 0.3 , 0.2 , 0.1 )
ϵ = 0	re c k	0.0297	0.0291	0.0317	0.0244
	re p k	0.0267	0.0267	0.0267	0.0267
	re k	0.0282	0.0279	0.0292	0.0256

ϵ = 0.001	re c k	0.0273	0.0271	0.0274	0.0245
	re p k	0.0269	0.0269	0.0270	0.0268
	re k	0.0271	0.0270	0.0272	0.0257

ϵ = 0.005	re c k	0.0402	0.0392	0.0521	0.0349
	re p k	0.0297	0.0296	0.0311	0.0294
	re k	0.0350	0.0344	0.0416	0.0322

ϵ = 0.01	re c k	0.0803	0.0688	0.1566	0.0575
	re p k	0.0374	0.0368	0.0416	0.0363
	re k	0.0588	0.0528	0.0991	0.0469

Example 2

Suppose the unknown time potential is

c ⁢ ( t ) = { t , t ∈ [ 0 , 1 / 3 ) , 1 / 3 , t ∈ [ 1 / 3 , 2 / 3 ) , 1 − t , t ∈ [ 1 / 3 , 1 ] ,

and the unknown time source term is

p ⁢ ( t ) = { t , t ∈ [ 0 , 0.5 ] , 1 − t , t ∈ ( 0.5 , 1 ] .

Taking the space source function r ⁢ ( x ) = cos ⁡ ( 2 ⁢ π ⁢ x ) and the initial state u 0 ⁢ ( x ) = cos ⁡ ( π ⁢ x ) , the additional measured data u ⁢ ( x 1 , t ) , u ⁢ ( x 2 , t ) are obtained by solving direct problem (1.1) by using the finite difference method.

Figure 3

The numerical results for Example 2 for α = ( 0.8 , 0.4 , 0.2 ) with regularization term.

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Figure 4

The numerical results for Example 2 for α = ( 0.8 , 0.4 , 0.2 ) without regularization term.

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Figure 5

The numerical results for Example 2 for α = ( 0.9 , 0.8 , 0.4 ) with regularization term.

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Figure 6

The numerical results for Example 2 for α = ( 0.9 , 0.8 , 0.4 ) without regularization term.

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Table 2

The relative errors of Example 2 for different orders 𝜶 and noise levels 𝜖.

𝜖 \ 𝜶	Error	( 0.9 , 0.5 , 0.1 )	( 0.9 , 0.3 , 0.2 )	( 0.8 , 0.7 , 0.6 )	( 0.3 , 0.2 , 0.1 )
ϵ = 0	re c k	0.0117	0.0116	0.0118	0.0113
	re p k	0.0278	0.0278	0.0278	0.0278
	re k	0.0198	0.0197	0.0198	0.0196

ϵ = 0.001	re c k	0.0218	0.0208	0.0261	0.0170
	re p k	0.0279	0.0279	0.0280	0.0279
	re k	0.0249	0.0244	0.0270	0.0224

ϵ = 0.005	re c k	0.0655	0.0650	0.0662	0.0557
	re p k	0.0301	0.0300	0.0312	0.0298
	re k	0.0478	0.0475	0.0487	0.0428

ϵ = 0.01	re c k	0.0798	0.0796	0.0864	0.0831
	re p k	0.0361	0.0357	0.0394	0.0352
	re k	0.0580	0.0576	0.0629	0.0592

The numerical results for Example 2 with α = ( 0.8 , 0.4 , 0.2 ) by using the discrepancy principle for various noise levels in the cases of μ , ν ≠ 0 and μ = ν = 0 are shown in Figures 3–4. We also choose the initial guess as ( c 0 , p 0 ) = ( c * , p * ) = ( 0 , 0 ) , the dimension N = 10 and M = 5 , the numerical differential step τ = 0.02 , and the regularization parameters chosen by μ = δ 3 / 2 , ν = 0.1 ⁢ δ 3 / 2 . The numerical results for Example 2 with α = ( 0.9 , 0.8 , 0.4 ) are displayed in Figures 5–6.

From Figures 3–6, we can easily find that the regularization term plays a very important role in the reconstruction of non-smooth functions. It is also verified that the algorithm is insensitive to fractional orders. From Table 2, we also find that our algorithm has a good reconstruction effect on non-smooth functions.

6.2 Two-Dimensional Case

Denote the space coordinates as ( x , y ) . The space domain Ω is taken as ( 0 , 1 ) × ( 0 , 1 ) , and take the measured points x 1 = { 0 , 0 } and x 2 = { 1 , 1 } in two-dimensional case. Let the elliptic operator be L = − Δ and the basis function spaces Φ k = span ⁡ { 2 ⁢ sin ⁡ ( π ⁢ t ) , … , 2 ⁢ sin ⁡ ( k ⁢ π ⁢ t ) } , k = M , N . Here we present the numerical results for one example to show the accuracy and stability of our proposed method.

Example 3

Suppose the unknown time potential is c ⁢ ( t ) = 17 ⁢ t ⁢ ( 1 − t ) and the time source is

p ⁢ ( t ) = 2 ⁢ sin ⁡ ( 2 ⁢ π ⁢ t ) + 3 ⁢ t ⁢ ( 1 − t ) 2 .

Taking the space source function r ⁢ ( x , y ) = cos ⁡ ( π ⁢ x ) ⁢ cos ⁡ ( π ⁢ y ) and the initial state u 0 ⁢ ( x , y ) = cos ⁡ ( π ⁢ x ) ⁢ cos ⁡ ( π ⁢ y ) , the additional measured data u ⁢ ( x 1 , t ) , u ⁢ ( x 2 , t ) are obtained by solving direct problem (1.1) by using the finite difference method.

Figure 7

The numerical results for Example 3 for α = ( 0.9 , 0.7 , 0.2 ) .

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

Figure 8

The numerical results for Example 3 for α = ( 0.8 , 0.5 , 0.3 ) .

(a)

c ⁢ ( t )

(b)

p ⁢ ( t )

The numerical results for Example 3 by using the discrepancy principle for various noise levels in the cases of α = ( 0.9 , 0.7 , 0.2 ) and α = ( 0.8 , 0.5 , 0.3 ) are shown in Figures 7–8. We choose the initial guess as

( c 0 , p 0 ) = ( c * , p * ) = ( 0 , 0 ) ,

the dimension N = 2 and M = 4 , the numerical differential step τ = 0.02 , and the regularization parameters chosen by μ = 0.002 ⁢ δ 3 / 2 , ν = 0.02 ⁢ δ 3 / 2 .

Table 3

The relative errors of Example 3 for different orders 𝜶 and noise levels 𝜖.

𝜖 \ 𝜶	Error	( 0.9 , 0.7 , 0.2 )	( 0.8 , 0.5 , 0.3 )	( 0.8 , 0.7 , 0.1 )
ϵ = 0	re c k	0.0486	0.0513	0.0411
	re p k	0.0183	0.0272	0.0213
	re k	0.0335	0.0393	0.0312

ϵ = 0.001	re c k	0.1265	0.1527	0.1484
	re p k	0.0287	0.0330	0.0317
	re k	0.0776	0.0929	0.0901

ϵ = 0.005	re c k	0.2051	0.2141	0.2075
	re p k	0.0457	0.0489	0.0463
	re k	0.1254	0.1315	0.1269

ϵ = 0.01	re c k	0.2013	0.2211	0.2100
	re p k	0.0471	0.0523	0.0486
	re k	0.1242	0.1367	0.1293

In Table 3, we choose α = ( 0.9 , 0.7 , 0.2 ) , ( 0.8 , 0.5 , 0.3 ) , ( 0.8 , 0.7 , 0.1 ) and show the numerical errors re c k , re p k , re k of Example 3 for different 𝜖. From Figures 7–8 and Table 3, we can see that the numerical results of the smooth potential and source for Example 3 match the exact ones quite well in the two-dimensional case even up to 1 % noise added in the “exact” Dirichlet data ( u ⁢ ( 0 , 0 , t ) , u ⁢ ( 1 , 1 , t ) ) . In addition, with fixed 𝜖, it is also observed that the fractional order has little effect on the accuracy of the numerical inversion results. With fixed 𝜶, we can see that the numerical error is decreasing as the level of noise becomes smaller. Finally, from the relative errors of reconstructed parameters c k ⁢ ( t ) and p k ⁢ ( t ) , it is easy to find that the inversion of the time source term is obviously better than the inversion of the potential function. This also shows that nonlinear inverse problems are more difficult than linear inverse problems.

7 Conclusions

We devote this paper to identifying the time-dependent potential term and time source simultaneously in a multi-term time-fractional diffusion equation. The existence, uniqueness and some regularities of the solution for the direct problem are obtained. Then the conditional stability of the inverse problem is provided by using the regularity of the corresponding direct problem, a nice property of the Caputo fractional derivative and some generalized Gronwall’s inequalities. Finally, we transform the inverse problem into a variational problem and employ the modified Levenberg–Marquardt method to find the approximation of the regularized solution. In the present paper, we can see that the traditional method of eigenfunction expansion is flawed (see, e.g., [40]) because the coefficients of the elliptic operator are not just depending on the spatial variable. Therefore, we apply the fixed point theorem to obtain the existence and uniqueness of the solution, and also obtain some nice regularity results to prove the conditional stability of the inverse problem.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12201502

Funding source: Science Fund for Distinguished Young Scholars of Gansu Province

Award Identifier / Grant number: 20JR10RA099

Funding source: Special Fund Project of Guiding Scientific and Technological Innovation Development of Gansu Province

Award Identifier / Grant number: 2020B-088

Funding statement: This work is supported by the NSF of China (grant no. 12201502), the Youth Science and Technology Fund of Gansu Province (grant no. 20JR10RA099) and the Innovation Capacity Improvement Project for Colleges and Universities of Gansu Province (grant no. 2020B-088).

References

[1] E. E. Adams and L. W. Gelhar, Field-study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resour. Res. 28 (1992), no. 12, 3293–3307. 10.1029/92WR01757Search in Google Scholar

[2] R. A. Adams, Sobolev spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar

[3] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res. 36 (2000), no. 6, 1403–1412. 10.1029/2000WR900031Search in Google Scholar

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar

[5] W. Bu, L. Ji, Y. Tang and J. Zhou, Space-time finite element method for the distributed-order time fractional reaction diffusion equations, Appl. Numer. Math. 152 (2020), 446–465. 10.1016/j.apnum.2019.11.010Search in Google Scholar

[6] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar

[7] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82–86. 10.3792/pja/1195521686Search in Google Scholar

[8] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal. 18 (2015), no. 3, 799–820. 10.1515/fca-2015-0048Search in Google Scholar

[9] R. Gorenflo and M. Yamamoto, Operator-theoretic treatment of linear Abel integral equations of first kind, Japan J. Indust. Appl. Math. 16 (1999), no. 1, 137–161. 10.1007/BF03167528Search in Google Scholar

[10] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems, Surveys Math. Indust. 3 (1993), no. 4, 253–315. Search in Google Scholar

[11] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resour. Res. 34 (1998), no. 5, 1027–1033. 10.1029/98WR00214Search in Google Scholar

[12] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. 10.1007/BFb0089647Search in Google Scholar

[13] D. Jiang and Z. Li, Coefficient inverse problem for variable order time-fractional diffusion equations from distributed data, Calcolo 59 (2022), no. 4, Paper No. 34. 10.1007/s10092-022-00476-3Search in Google Scholar

[14] S. Z. Jiang and Y. J. Wu, Recovering a time-dependent potential function in a multi-term time fractional diffusion equation by using a nonlinear condition, J. Inverse Ill-Posed Probl. 29 (2021), no. 2, 233–248. 10.1515/jiip-2019-0055Search in Google Scholar

[15] B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), 825–843. 10.1016/j.jcp.2014.10.051Search in Google Scholar

[16] B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28 (2012), no. 7, Article ID 075010. 10.1088/0266-5611/28/7/075010Search in Google Scholar

[17] X. Jing and J. Peng, Simultaneous uniqueness for an inverse problem in a time-fractional diffusion equation, Appl. Math. Lett. 109 (2020), Article ID 106558. 10.1016/j.aml.2020.106558Search in Google Scholar

[18] B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems 35 (2019), no. 6, Article ID 065004. 10.1088/1361-6420/ab109eSearch in Google Scholar

[19] Y. Kian and M. Yamamoto, Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations, Fract. Calc. Appl. Anal. 24 (2021), no. 1, 168–201. 10.1515/fca-2021-0008Search in Google Scholar

[20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006. Search in Google Scholar

[21] A. Kubica and M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal. 21 (2018), no. 2, 276–311. 10.1515/fca-2018-0018Search in Google Scholar

[22] K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2 (1944), 164–168. 10.1090/qam/10666Search in Google Scholar

[23] M. Levy and B. Berkowitz, Measurement and analysis of non-fickian dispersion in heterogeneous porous media, J. Contam. Hydrol. 64 (2003), no. 3, 203–226. 10.1016/S0169-7722(02)00204-8Search in Google Scholar PubMed

[24] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2108–2131. 10.1137/080718942Search in Google Scholar

[25] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems 32 (2016), no. 1, Article ID 015004. 10.1088/0266-5611/32/1/015004Search in Google Scholar

[26] Z. Li, Y. Liu and M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput. 257 (2015), 381–397. 10.1016/j.amc.2014.11.073Search in Google Scholar

[27] Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem, Comput. Math. Appl. 73 (2017), no. 1, 96–108. 10.1016/j.camwa.2016.10.021Search in Google Scholar

[28] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fract. Calc. Appl. Anal. 19 (2016), no. 4, 888–906. 10.1515/fca-2016-0048Search in Google Scholar

[29] Y. Luchko, Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl. 374 (2011), no. 2, 538–548. 10.1016/j.jmaa.2010.08.048Search in Google Scholar

[30] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam. 24 (1999), no. 2, 207–233. Search in Google Scholar

[31] F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo, Time-fractional diffusion of distributed order, J. Vib. Control 14 (2008), no. 9–10, 1267–1290. 10.1177/1077546307087452Search in Google Scholar

[32] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math. 11 (1963), 431–441. 10.1137/0111030Search in Google Scholar

[33] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1–77. 10.1016/S0370-1573(00)00070-3Search in Google Scholar

[34] R. Metzler, J. Klafter and I. M. Sokolov, Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended, Phys. Rev. E 58 (1998), no. 2, 1621–1633. 10.1103/PhysRevE.58.1621Search in Google Scholar

[35] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems 29 (2013), no. 7, Article ID 075013. 10.1088/0266-5611/29/7/075013Search in Google Scholar

[36] V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York, 1984. 10.1007/978-1-4612-5280-1Search in Google Scholar

[37] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl. 56 (2008), no. 4, 1138–1145. 10.1016/j.camwa.2008.02.015Search in Google Scholar

[38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. 10.1007/978-1-4612-5561-1Search in Google Scholar

[39] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[40] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[41] R. Schumer, D. A. Benson, M. M. Meerschaert and B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res. 39 (2003), no. 10, Article ID 1296. 10.1029/2003WR002141Search in Google Scholar

[42] M. Stynes, E. O’Riordan and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079. 10.1137/16M1082329Search in Google Scholar

[43] L. Sun and T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation, Appl. Numer. Math. 111 (2017), 160–180. 10.1016/j.apnum.2016.09.005Search in Google Scholar

[44] L. Sun, Y. Zhang and T. Wei, Recovering the time-dependent potential function in a multi-term time-fractional diffusion equation, Appl. Numer. Math. 135 (2019), 228–245. 10.1016/j.apnum.2018.09.001Search in Google Scholar

[45] L. L. Sun, Y. S. Li and Y. Zhang, Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation, Inverse Problems 37 (2021), no. 5, Article ID 055007. 10.1088/1361-6420/abf162Search in Google Scholar

[46] V. K. Tuan, Inverse problem for fractional diffusion equation, Fract. Calc. Appl. Anal. 14 (2011), no. 1, 31–55. 10.2478/s13540-011-0004-xSearch in Google Scholar

[47] T. Wei and K. Liao, Identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point, Appl. Anal. 101 (2022), no. 18, 6522–6547. 10.1080/00036811.2021.1932834Search in Google Scholar

[48] T. Wei and J. Xian, Determining a time-dependent coefficient in a time-fractional diffusion-wave equation with the Caputo derivative by an additional integral condition, J. Comput. Appl. Math. 404 (2022), Paper No. 113910. 10.1016/j.cam.2021.113910Search in Google Scholar

[49] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems 28 (2012), no. 10, Article ID 105010. 10.1088/0266-5611/28/10/105010Search in Google Scholar

[50] X.-B. Yan, Z.-Q. Zhang and T. Wei, Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation, Chaos Solitons Fractals 157 (2022), Paper No. 111901. 10.1016/j.chaos.2022.111901Search in Google Scholar

[51] H. Ye, F. Liu and V. Anh, Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys. 298 (2015), 652–660. 10.1016/j.jcp.2015.06.025Search in Google Scholar

[52] Z. Zhang, An undetermined coefficient problem for a fractional diffusion equation, Inverse Problems 32 (2016), no. 1, Article ID 015011. 10.1088/0266-5611/32/1/015011Search in Google Scholar

[53] Z. Zhang, An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Probl. Imaging 11 (2017), no. 5, 875–900. 10.3934/ipi.2017041Search in Google Scholar

[54] Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math. 82 (2017), no. 3, 579–600. 10.1093/imamat/hxx004Search in Google Scholar

[55] M. Zheng, F. Liu, V. Anh and I. Turner, A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model. 40 (2016), no. 7–8, 4970–4985. 10.1016/j.apm.2015.12.011Search in Google Scholar

Received: 2022-10-20

Revised: 2023-02-27

Accepted: 2023-05-03

Published Online: 2023-07-12

Published in Print: 2024-01-01

Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation

Abstract

1 Introduction

2 Preliminaries

Proposition 2.2 ([26])

Lemma 2.3 ([4])

Lemma 2.4 (see [47, Lemma 2.7])

3 Existence, Uniqueness and Regularity of the Solution for the Direct Problem

Proof

Proof of Theorem 3.1

4 Stability of the Inverse Problem

Proof

5 Ill-Posedness and Regularization Method

Proof

Lemma 5.2 ([4])

Proof

Proof

6 Numerical Experiments

6.1 One-Dimensional Case

6.2 Two-Dimensional Case

7 Conclusions

References

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