A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics

Abstract

The inverse problem of reconstructing the right-hand side (RHS) of a mixed problem for one-dimensional diffusion equation with variable space operator is considered. The well-posedness of this problem in Hölder spaces is established.

1. Introduction

It is known that many applied problems in fluid mechanics, other areas of physics, and mathematical biology were formulated as the mathematical model of partial differential equations of the variable types [1–3]. A model for transport across microvessel endothelium was developed to determine the forces and bending moments acting on the structure of the flow over endothelial cells (ECs) [4]. Computational blood flow analysis through glycocalyx on the EC is performed as a direct problem previously under smooth and nonsmooth initial conditions (see [5–7]). But it is known that, due to the lack of some data and/or coefficients, many real-life problems are modeled as inverse problems [8–11].

In this paper, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation arisen in computational blood flow analysis is investigated. The importance of well-posedness has been widely recognized by the researchers in the field of partial differential equations [12–16]. Moreover, the well-posedness of the RHS identification problems for a parabolic equation where the unknown function 𝑝 is in space variable and in time variable is well investigated [17–27]. As it is known, well-posedness in the sense of Hadamard means that there is existence and uniqueness of the solution and the solution is stable. In this study, we deal with the stability analysis of the inverse problem of reconstructing the right-hand side. The existence of a solution for two-phase flow in porous media has been studied previously (for instance, see [28]).

1.1. Problem Formulation

Blood flow over the EC inside the arteries is modeled in two regions (see [6]). Core region (0<𝑥<𝑙) flow is defined through the center of capillary and porous region (𝑙<𝑥<𝐿) flow is through the glycocalyx. RHS function includes the pressure difference along the microchannels under unsteady fluid flow conditions. When the pressure difference is an unknown function of 𝑡, we reach a new model, and, by overdetermined (additional) conditions derived from an observation point, the solution of this problem can be obtained. The model can be considered as the mixed problem for one-dimensional diffusion equation with variable space operator: 𝜕𝑢(𝑡,𝑥)(𝑥)𝜕𝜕𝑡=𝑎²𝑢(𝑡,𝑥)𝜕𝑥²(𝑡)𝑞(𝑥)()()(],()+𝑝+𝑓𝑡,𝑥,𝑥∈0,𝑙,𝑡∈0,𝑇𝜕𝑢𝑡,𝑥(𝑥)𝜕𝜕𝑡=𝑎²𝑢(𝑡,𝑥)𝜕𝑥²()𝑢()(𝑡)𝑞(𝑥)()()(],𝑢()(𝑥)[],𝑢+𝑏𝑡,𝑥𝑡,𝑥+𝑝+𝑔𝑡,𝑥,𝑥∈𝑙,𝐿,𝑡∈0,𝑇0,𝑥=𝜑,𝑥∈0,𝐿_𝑥()()[],𝑢()()𝑡,0=0,𝑢𝑡,𝐿=0,𝑡∈0,𝑇𝑡,𝑙+=𝑢𝑡,𝑙−,𝑢_𝑥(𝑡,l+)=𝑢_𝑥()[],𝑢𝑡,𝑙−,𝑡∈0,𝑇𝑡,𝑥^∗=𝜌(𝑡),0≤𝑥^∗≤𝑙,0≤𝑡≤𝑇. (1.1) Here, 𝑢(𝑡,𝑥) and 𝑝(𝑡) are unknown functions, 𝑎(𝑥),   𝑏(𝑡,𝑥),𝑓(𝑡,𝑥),𝑔(𝑡,𝑥),𝜌(𝑡), and 𝜑(𝑥) are given sufficiently smooth functions, and 𝑎(𝑥)⩾𝑎>0. Also, 𝑞(𝑥) is a sufficiently smooth function assuming that 𝑞^′(0)=𝑞(𝐿)=0 and 𝑞(𝑥^∗)≠0.

2. Main Results

2.1. Differential Case

To formulate our results, we introduce the Banach space 𝐶^∘𝛼[0,𝐿],   𝛼∈(0,1), of all continuous functions 𝜙(𝑥) defined on [0,𝐿] with 𝜙^′(0)=𝜙(𝐿)=0 satisfying a Hölder condition for which the following norm is finite: 𝜙^∘_𝐶^𝛼_[0,𝐿]=max_{0≤𝑥≤𝐿}𝜙(𝑥)+sup_{0≤𝑥<𝑥+ℎ≤𝐿}𝜙(𝑥+ℎ)−𝜙(𝑥)ℎ^𝛼. (2.1)

In a Banach space 𝐸, with the help of a positive operator 𝐴 we introduce the fractional spaces 𝐸_𝛼,0<𝛼<1, consisting of all 𝑣∈𝐸 for which the following norm is finite: ‖𝑣‖_𝐸𝛼=‖𝑣‖_𝐸+sup_𝜆>0𝜆^1−𝛼𝐴exp{−𝜆𝐴}𝑣_𝐸. (2.2) Positive constants will be indicated by 𝑀 which can be differ in time. On the other hand 𝑀_𝑖(𝛼,𝛽,…) is used to focus on the fact that the constant depends only on 𝛼,𝛽,…, and the subindex 𝑖 is used to indicate a different constant.

Theorem 2.1. Let 𝜑∈𝐶^∘2𝛼+2[0,𝐿],  𝐹₁∈𝐶([0,𝑇],𝐶^∘2𝛼[0,𝐿]), and 𝜌^′∈𝐶[0,𝑇]. Then for the solution of problem (1.1), the following coercive stability estimates 𝑢_{𝑡𝐶([0,𝑇],∘𝐶}^2𝛼_[0,𝐿])+‖𝑢‖_{𝐶([0,𝑇],∘𝐶}^2𝛼+2_[0,𝐿])≤𝑀𝑥^∗,𝑞𝜌^′_𝐶[0,𝑇]+𝑀𝑎,𝛿,𝜎,𝛼,𝑥^∗×𝜑,𝑞,𝑇_{∘𝐶2𝛼+2[0,𝐿]}+𝐹_{1𝐶([0,𝑇],∘𝐶}^2𝛼_[0,𝐿])+𝜌_𝐶[0,𝑇],𝑝_𝐶[0,𝑇]≤𝑀𝑥^∗,𝑞𝜌^′_𝐶[0,𝑇]+𝑀𝑎,𝛿,𝜎,𝛼,𝑥^∗,𝑞,𝑇𝜑_∘𝐶^2𝛼+2_[0,𝐿]+𝐹_{1𝐶([0,𝑇],∘𝐶}^2𝛼_[0,𝐿])+𝜌_𝐶[0,𝑇] (2.3) hold.

Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [23]): 𝑢(𝑡,𝑥)=𝜂(𝑡)𝑞(𝑥)+𝑤(𝑡,𝑥), (2.4) where 𝜂(𝑡)=^𝑡₀𝑝(𝑠)𝑑𝑠. (2.5) Taking derivatives from (2.4) with respect to 𝑡 and 𝑥, we get 𝜕𝑢(𝑡,𝑥)(𝑡)𝑞(𝑥)+()𝜕𝑡=𝑝𝜕𝑤𝑡,𝑥,𝜕𝜕𝑡₂𝑢(𝑡,𝑥)𝜕𝑥²(𝑡)𝑑=𝜂²𝑞(𝑥)𝑑𝑥²+𝜕²𝑤(𝑡,𝑥)𝜕𝑥². (2.6) Moreover, substituting 𝑥 by 𝑥^∗ in (2.4), we obtain 𝑢𝑡,𝑥^∗=𝜂(𝑡)𝑞𝑥^∗+𝑤𝑡,𝑥^∗(𝑡),𝜂(𝑡)=𝜌(𝑡)=𝜌−𝑤𝑡,𝑥^∗𝑞(𝑥^∗). (2.7) Differentiating both sides of (2.7) with respect to 𝑡, we get 𝑝(𝑡)=𝜌^′(𝑡)−𝑤_𝑡𝑡,𝑥^∗𝑞(𝑥^∗). (2.8) From identity (2.8) and the triangle inequality, it follows that 𝑝(𝑡=𝜌)^′(𝑡)−𝑤_𝑡𝑡,𝑥^∗𝑞(𝑥^∗)≤𝑀𝑥^∗,𝑞𝜌^′(𝑡)+𝑤_𝑡𝑡,𝑥^∗≤𝑀𝑥^∗,𝑞max_{0≤𝑡≤𝑇}𝜌^′(𝑡)+max_{0≤𝑡≤𝑇}max_{0≤𝑥≤𝐿}𝑤_𝑡(𝑥𝑡,𝑥)≤𝑀^∗,𝑞max_{0≤𝑡≤𝑇}𝜌^′(𝑡)+max_{0≤𝑡≤𝑇}𝑤_𝑡(𝑡)_∘𝐶^2𝛼_[0,𝐿], (2.9) for any 𝑡,𝑡∈[0,𝑇]. Using problem (1.1) and (2.4)–(2.7), one can show that 𝑤(𝑡,𝑥) is the solution of the following problem: 𝜕𝑤(𝑡,𝑥)(𝑥)𝜕𝜕𝑡=𝑎²𝑤(𝑡,𝑥)𝜕𝑥²(𝑥)𝜌(𝑡)+𝑎−𝑤𝑡,𝑥^∗𝑞(𝑥^∗)𝑑²𝑞(𝑥)𝑑𝑥²()()(],()+𝑓𝑡,𝑥,𝑥∈0,𝑙,𝑡∈0,𝑇𝜕𝑤𝑡,𝑥(𝑥)𝜕𝜕𝑡=𝑎²𝑤(𝑡,𝑥)𝜕𝑥²()𝑤()+𝜌(𝑡)+𝑏𝑡,𝑥𝑡,𝑥−𝑤𝑡,𝑥^∗𝑞(𝑥^∗)𝑎(𝑥)𝑑²𝑞(𝑥)𝑑𝑥²()𝑞(𝑥)()()(],𝑤()(𝑥)[],𝑤+𝑏𝑡,𝑥+𝑔𝑡,𝑥,𝑥∈𝑙,𝐿,𝑡∈0,𝑇0,𝑥=𝜑,𝑥∈0,𝐿_𝑥()()[],𝑤()()𝑡,0=0,𝑤𝑡,𝐿=0,𝑡∈0,𝑇𝑡,𝑙+=𝑤𝑡,𝑙−,𝑤_𝑥(𝑡,𝑙+)=𝑤_𝑥(𝑡,𝑙−),𝑡∈[0,𝑇], (2.10) under the same assumptions on 𝑞(𝑥). Estimate (2.9) and the following theorem conclude the proof of Theorem 2.1.

Theorem 2.2. For the solution of problem (2.10), the following coercive stability estimate 𝑤_𝑡∘𝐶^2𝛼_[0,𝐿]≤𝑀𝑎,𝛿,𝜎,𝛼,𝑥^∗×𝜑,𝑞,𝑇_{∘𝐶2𝛼+2[0,𝐿]}+𝐹_{1𝐶([0,𝑇],∘𝐶}^2𝛼_[0,𝐿])+𝜌_𝐶[0,𝑇] (2.11) holds.

Proof. Let us rewrite problem (2.10) in the abstract form as an initial-value problem: 𝑤_𝑡+𝐴𝑤+𝐵𝑤=𝑎𝑞^′′𝜌(𝑡)−𝜎𝑞−𝑤𝑡,𝑥^∗𝑞^∗+𝐹₁(𝑡)+𝐹₂(𝑡)𝑤(0),0<𝑡≤𝑇,=𝜑 (2.12) in the Banach space 𝐸=𝐶^∘[0,𝐿]. Here, the positive operator 𝐴 is defined by (𝑥)𝜕𝐴𝑢=−𝑎²𝑢(𝑡,𝑥)𝜕𝑥²+𝜎𝑢, (2.13) with 𝐷(𝐴)=𝑢(𝑥)∶𝑢,𝑢^′,𝑢^′′∈𝐶[0,𝐿],𝑢_𝑥(0)=𝑢(𝐿)=0, (2.14) and for every fixed 𝑡∈[0,𝑇], the differential operator 𝐵 is given by the formula 𝐵(𝑡)𝑢=−𝜎𝑢^𝑛−((𝑡(𝑡)(),0≤𝑥<𝑙,𝜎−𝑏))𝑢,𝑏=𝑏𝑡,𝑥,𝑙<𝑥≤𝐿. (2.15) Here, 𝜎 is a positive constant. The right-hand side functions are defined by 𝐹₁(𝑡)=𝑓(𝑡)𝑔(𝑡)+𝜌(𝑡),0≤𝑥<𝑙,0,𝑞^∗𝑏(𝑡)𝐹𝑞,𝑙<𝑥≤𝐿,₂(𝑡)=−𝑤0,0≤𝑥<𝑙,0,𝑡,𝑥^∗𝑞^∗𝑏(𝑡)𝑞,𝑙<𝑥≤𝐿, (2.16) where 𝑓(𝑡)=𝑓(𝑡,𝑥),𝑔(𝑡)=𝑔(𝑡,𝑥),𝑏(𝑡)=𝑏(𝑡,𝑥) are known, and 𝑤(𝑡)=𝑤(𝑡,𝑥) is unknown abstract functions defined on [0,𝑇] with values in 𝐸=𝐶^∘[0,𝐿],    𝑤(𝑡,𝑥^∗) is unknown scalar function defined on [0,𝑇],   𝑞=𝑞(𝑥),   𝑞^′′=𝑞^′′(𝑥),   𝜑=𝜑(𝑥), and  𝑎=𝑎(𝑥) are elements of 𝐸=𝐶^∘[0,𝐿], and 𝑞^∗=𝑞(𝑥^∗) is a number.
It is known that operator-𝐴 generates an analytic semigroup exp{−𝑡𝐴}(𝑡>0) and the following estimate holds: 𝐴^𝛼exp{−𝑡𝐴}_𝐸→𝐸≤𝑀𝑒^−𝛿𝑡𝑡^−𝛼,0≤𝛼≤1, (2.17) where 𝑡,𝛿,𝑀>0 [29].
By the Cauchy formula, the solution can be written as 𝑤(𝑡)=𝑒^−𝑡𝐴𝜑−^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑤𝑠,𝑥^∗𝑑𝑠+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝜌(𝑠)𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑑𝑠+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)+𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝐹₂(𝑠)𝑑𝑠−^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝑤(𝑠)𝑑𝑠. (2.18) Then, the following presentation of the solution of abstract problem (2.12) exists: 𝐴𝑤(𝑡)=𝐴𝑒^−𝑡𝐴𝜑−^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑤𝑠,𝑥^∗𝑑𝑠+^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝜌(𝑠)𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑑𝑠+^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)+𝑑𝑠_𝑡0𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₂(𝑠)𝑑𝑠+^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝑤(𝑠)𝑑𝑠=⁶_𝑘=1𝐺_𝑘(𝑡). (2.19) Here, 𝐺₁(𝑡)=𝐴𝑒^−𝑡𝐴𝐺𝜑,₂(𝑡)=−^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑤𝑠,𝑥^∗𝐺𝑑𝑠,₃(𝑡)=−^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝜌(𝑠)𝑎𝑞^′′−𝜎𝑞𝑞^∗𝐺𝑑𝑠,₄(𝑡)=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝐺𝑑𝑠,₅(𝑡)=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₂(𝑠)𝐺𝑑𝑠,₆(𝑡)=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝑤(𝑠)𝑑𝑠. (2.20) From the fact that the operators 𝑅,exp{−𝜆𝐴} and 𝐴 commute, it follows that [29] ‖𝑅‖_{𝐸𝛼→𝐸𝛼}≤‖𝑅‖_𝐸→𝐸. (2.21) Now, we estimate 𝐺_𝑘(𝑡) for 𝑘=1,2,…,5 separately. Applying the definition of norm of the spaces 𝐸_𝛼 and estimate (2.21), we get 𝐺₁(𝑡)_𝐸𝛼=𝐴𝑒^−𝑡𝐴𝜑_𝐸𝛼≤𝑒^−𝑡𝐴_{𝐸𝛼→𝐸𝛼}𝐴𝜑_𝐸𝛼≤𝑒^−𝑡𝐴_𝐸→𝐸𝐴𝜑_𝐸𝛼. (2.22) Then, using estimate (2.17) for 𝛼=0, we reach to 𝐺₁(𝑡)_𝐸𝛼≤𝑀₁𝐴𝜑_𝐸𝛼, (2.23) for any 𝑡,   𝑡∈[0,𝑇].
Let us estimate 𝐺₂(𝑡): 𝐺₂(𝑡)_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑤𝑠,𝑥^∗≤𝑑𝑠_𝑡0𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝑤𝑠,𝑥^∗𝑑𝑠. (2.24) By the definition of norm of the spaces 𝐸_𝛼, we have that ^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝑑𝑠=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝑑𝑠+sup_𝜆>0^𝑡₀𝜆^1−𝛼𝐴𝑒^−𝜆𝐴𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝑑𝑠. (2.25) Let us estimate the first term. From the definition of norm of the spaces 𝐸_𝛼 it follows that ^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝑑𝑠=^𝑡₀(𝑡−𝑠)^𝛼−1(𝑡−𝑠)^1−𝛼𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑑𝑠𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼≤𝑇^𝛼𝛼𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼=𝑀₂𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇. (2.26) Using estimate (2.17), we obtain ^𝑡₀𝜆^1−𝛼𝐴𝑒^−𝜆𝐴𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝑑𝑠≤^𝑡₀2^2−𝛼𝜆^1−𝛼(𝜆+𝑡−𝑠)^2−𝛼𝑑𝑠𝜆+𝑡−𝑠2𝐴𝑒^{−((𝜆+𝑡−𝑠)/2)𝐴}_𝐸→𝐸×𝜆+𝑡−𝑠2^1−𝛼𝐴𝑒^{−((𝜆+𝑡−𝑠)/2)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸≤𝑀₃(𝛼)𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼^𝑡₀𝜆^1−𝛼(𝜆+𝑡−𝑠)^2−𝛼𝑑𝑠≤𝑀₃(𝛼)𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝜆^1−𝛼(1−𝛼)(𝜆+𝑡)^1−𝛼, (2.27) for any 𝜆>0. From that it follows sup_𝜆>0^𝑡₀𝜆^1−𝛼𝐴𝑒^−𝜆𝐴𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝑑𝑠≤𝑀₃(𝛼)𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼1(1−𝛼)=𝑀₄𝑎,𝜎,𝛼,𝑥^∗,𝑞. (2.28) Then, we get ^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝑑𝑠≤𝑀₅𝑎,𝜎,𝛼,𝑥^∗𝐺,𝑞,𝑇₂(𝑡)_𝐸𝛼≤𝑀₆𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇^𝑡₀𝑤𝑠,𝑥^∗𝑑𝑠. (2.29)(2.30) Using definitions of norm of spaces 𝐸 and 𝐸_𝛼 and estimate (2.21), we obtain that 𝑤𝑠,𝑥^∗≤‖𝑤‖_𝐸≤‖𝑤‖_𝐸𝛼=𝐴⁻¹𝐴𝑤_𝐸𝛼≤𝐴⁻¹_𝐸→𝐸‖𝐴𝑤‖_𝐸𝛼≤𝑀‖𝐴𝑤‖_𝐸𝛼,𝐺₂(𝑡)_𝐸𝛼≤𝑀₇𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇‖𝐴𝑤‖_𝐸𝛼, (2.31)(2.32) for any 𝑡∈[0,𝑇].
From estimate (2.29), the estimate of 𝐺₃(𝑡) is as follows: 𝐺₃(𝑡)_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝜌(𝑠)𝑎𝑞^′′−𝜎𝑞𝑞^∗𝑑𝑠_𝐸𝛼≤^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝑑𝑠𝜌_𝐶[0,𝑇]≤𝑀₈𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇𝜌_𝐶[0,𝑇]. (2.33) Now, let us estimate 𝐺₄(𝑡). By the definition of the norm of the spaces 𝐸_𝛼, we get 𝐺₄(𝑡)_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸+sup_𝜆>0𝜆^1−𝛼𝐴𝑒^{−𝜆𝐴𝑡}₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸. (2.34) Equation (2.2) yields that ^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸=^𝑡₀(𝑡−𝑠)^𝛼−1(𝑡−𝑠)^1−𝛼𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)_𝐸≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝑑𝑠𝐹_{1𝐶(𝐸𝛼)}=𝑡^𝛼𝛼𝐹_{1𝐶(𝐸𝛼)}≤𝑀₉(𝛼,𝑇)𝐹_{1𝐶(𝐸𝛼)}. (2.35) Now, we consider the second term. Using (2.2), we get 𝜆^1−𝛼𝐴𝑒^{−𝜆𝐴𝑡}₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸≤𝜆^1−𝛼𝑡₀𝑡−𝑠+𝜆2^𝛼−1𝑡−𝑠+𝜆2⁻¹×𝑡−𝑠+𝜆2𝐴𝑒^{−((𝑡−𝑠+𝜆)/2)𝐴}_𝐸→𝐸×𝑡−𝑠+𝜆2^1−𝛼𝐴𝑒^{−((𝑡−𝑠+𝜆)/2)𝐴}𝐹₁(𝑠)_𝐸𝑑𝑠≤𝑀₁₀𝜆^1−𝛼𝑡₀𝑡−𝑠+𝜆2^𝛼−2𝐹_1𝐸𝛼𝑑𝑠≤𝑀₁₁𝜆^1−𝛼𝑡₀𝑡−𝑠+𝜆2^𝛼−2𝑑𝑠𝐹_{1𝐶(𝐸𝛼)}, (2.36) for any 𝜆>0. Then, sup_𝜆>0𝜆^1−𝛼𝐴𝑒^{−𝜆𝐴𝑡}₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₁(𝑠)𝑑𝑠_𝐸≤𝑀₉2^1−𝛼𝐹1−𝛼^{1𝐶(𝐸𝛼)}=𝑀₁₁(𝛼)𝐹_{1𝐶(𝐸𝛼)}. (2.37) Combining estimates (2.35) and (2.37), we obtain 𝐺₄(𝑡)_𝐸𝛼≤𝑀₁₂(𝛼,𝑇)𝐹_{1𝐶(𝐸𝛼)}. (2.38) The estimate of 𝐺₅(𝑡) is as follows. Since operators 𝐴 and 𝑒^−𝑡𝐴 commute, we can write that 𝐺₅(𝑡)_𝐸𝛼≤^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐹₂(𝑠)𝑑𝑠_𝐸𝛼≤𝑀₁₃(𝛼,𝑇)‖𝐴𝑤‖_𝐸𝛼. (2.39) Let us estimate 𝐺₆(𝑡)∶𝐺₆(𝑡)_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝑤(𝑠)𝑑𝑠_𝐸𝛼=^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝐴⁻¹𝐴𝑤(𝑠)𝑑𝑠_𝐸𝛼≤^𝑡₀𝐴𝑒^{−(𝑡−𝑠)𝐴}𝐵(𝑠)𝐴⁻¹_{𝐸𝛼→𝐸𝛼}‖𝐴𝑤(𝑠)‖_𝐸𝛼𝑑𝑠. (2.40) Since 𝑒^−𝑡𝐴_{𝐸𝛼→𝐸𝛼}≤𝑒^−𝑡𝐴_𝐸→𝐸≤𝑀𝑒^−𝛿𝑡,𝐴𝐵(𝑠)𝐴⁻¹_{𝐸𝛼→𝐸𝛼}≤𝑀, (2.41) we get 𝐺₆(𝑡)_𝐸𝛼≤𝑀₁₄^𝑡₀‖𝐴𝑤(𝑠)‖_𝐸𝛼𝑑𝑠. (2.42) Finally combining estimates (2.23), (2.32), (2.33), (2.38), (2.39), and (2.42), we get ‖𝐴𝑤‖_𝐸𝛼≤𝑀₁𝐴𝜑_𝐸𝛼+𝑀₈𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇𝜌_𝐶[0,𝑇]+𝑀₁₂(𝛼,𝑇)𝐹_{1𝐶(𝐸𝛼)}+𝑀₁₅^𝑡₀‖𝐴𝑤(𝑠)‖_𝐸𝛼𝑑𝑠, (2.43) where 𝑀₁₅=𝑀₇+𝑀₁₃+𝑀₁₄.
Using Gronwall's inequality, we can write ‖𝐴𝑤‖_𝐸𝛼≤𝑒^𝑀15𝑇𝑀₁𝐴𝜑_𝐸𝛼+𝑀₈𝑎,𝜎,𝛼,𝑥^∗,𝑞,𝑇𝜌_𝐶[0,𝑇]+𝑀₁₂(𝛼,𝑇)𝐹_{1𝐶(𝐸𝛼)}. (2.44) Applying the formulas 𝑤𝑡,𝑥^∗=𝑤0,𝑥^∗+^𝑡₀𝑤_𝑧𝑧,𝑥^∗𝑑𝑧=𝜑𝑥^∗+^𝑡₀𝑤_𝑧𝑧,𝑥^∗𝜑𝑥𝑑𝑧,_∗≤max_{0≤𝑥≤𝐿}𝜑(𝑥)=𝜑_𝐸≤𝜑_𝐸𝛼≤𝐴⁻¹_{𝐸𝛼→𝐸𝛼}𝐴𝜑_𝐸𝛼≤𝑀𝐴𝜑_𝐸𝛼 (2.45) and the triangle inequality, we can write 𝑎𝑞^′′−𝜎𝑞𝑞^∗𝜌(𝑡)−𝑤𝑡,𝑥^∗_𝐸𝛼≤𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝜌_𝐶[0,𝑇]+𝑀₁₃𝐴𝜑_𝐸𝛼+^𝑡₀𝑤_𝑧𝐸𝛼𝑑𝑧. (2.46) Using boundedness of 𝐵, problem (2.12), and estimate (2.46), we have 𝑤_𝑡𝐸𝛼≤‖𝐴𝑤‖_𝐸𝛼+‖𝐵𝑤‖_𝐸𝛼+𝐹_{1𝐶(𝐸𝛼)}+𝑎𝑞^′′−𝜎𝑞𝑞^∗_𝐸𝛼𝜌_𝐶[0,𝑇]+𝑀₁₃𝐴𝜑_𝐸𝛼+^𝑡₀𝑤_𝑧𝐸𝛼𝑑𝑧. (2.47) So, Gronwall's inequality and the following theorem finish the proof of Theorem 2.3.

Theorem 2.3 (see [29]). For 0<𝛼<1/2, the spaces 𝐸_𝛼(𝐶[0,𝐿],𝐴) and 𝐶^2𝛼[0,𝐿] coincide and their norms are equivalent.

2.2. Difference Case

For the approximate solution of problem (1.1), the Rothe difference scheme 𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏=𝑎𝑥_𝑛𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝑝^𝑘𝑞_𝑛+𝑓𝑡_𝑘,𝑥_𝑛,1≤𝑘≤𝑁,1≤𝑛≤𝑀_𝑙−1,𝑀_𝑙𝑢ℎ=𝑙,𝑁𝜏=𝑇,_𝑘𝑛−𝑢_𝑛^𝑘−1𝜏=𝑎𝑥_𝑛𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝑏𝑡_𝑘,𝑥_𝑛𝑢^𝑘_𝑛+𝑝^𝑘𝑞_𝑛+𝑔𝑡_𝑘,𝑥_𝑛,1≤𝑘≤𝑁,𝑀_𝑙𝑢+1≤𝑛≤𝑀,𝑀ℎ=𝐿,𝑁𝜏=𝑇,_0𝑛=𝜑𝑥_𝑛𝑢,0≤𝑛≤𝑀,_𝑘1−𝑢^𝑘₀=𝑢^𝑘_𝑀𝑢=0,0≤𝑘≤𝑁,_{𝑘𝑀𝑙+1}−𝑢^𝑘_𝑀𝑙=𝑢^𝑘_𝑀𝑙−𝑢^𝑘_𝑀𝑙−1𝑢,0≤𝑘≤𝑁,_{𝑘⌊𝑥∗⌋/ℎ}=𝑢^𝑘_𝑠=𝜌𝑡_𝑘,0≤𝑘≤𝑁,0≤𝑠≤M, (2.48) where 𝑝^𝑘=𝑝(𝑡_𝑘),𝑞_𝑛=𝑞(𝑥_𝑛),𝑥_𝑛=𝑛ℎ, and 𝑡_𝑘=𝑘𝜏 is constructed. Here, 𝑞_𝑠≠0 and 𝑞₁−𝑞₀=𝑞_𝑀=0 are assumed. ⌊𝑥⌋ represents the floor function of 𝑥.

With the help of a positive operator 𝐴, we introduce the fractional spaces 𝐸^′_𝛼,0<𝛼<1, consisting of all 𝑣∈𝐸 for which the following norm is finite: ‖𝑣‖_𝐸′𝛼=‖𝑣‖_𝐸+sup_𝜆>0𝜆^𝛼𝐴(𝜆+𝐴)⁻¹𝑣_𝐸. (2.49) To formulate our results, we introduce the Banach space 𝐶^∘_ℎ^𝛼=𝐶^∘𝛼[0,𝐿]_ℎ,  𝛼∈(0,1), of all grid functions 𝜙^ℎ={𝜙_𝑛}^𝑀−1_𝑛=1 defined on [0,𝐿]_ℎ=𝑥_𝑛=𝑛ℎ,0≤𝑛≤𝑀,𝑀ℎ=𝐿, (2.50) with 𝜙₁−𝜙₀=𝜙_𝑀=0 equipped with the norm 𝜙_ℎ^∘_𝐶^𝛼_ℎ=𝜙_ℎ𝐶ℎ+sup_{1≤𝑛<𝑛+𝑟≤𝑀}𝜙_𝑛+𝑟−𝜙_𝑛(𝑟ℎ)^−𝛼,𝜙_ℎ𝐶ℎ=max_{1≤𝑛≤𝑀}𝜙_𝑛. (2.51) Moreover, 𝐶_𝜏(𝐸)=𝐶([0,T]_𝜏,𝐸) is the Banach space of all grid functions 𝜙^𝜏={𝜙(𝑡_𝑘)}^𝑁−1_𝑘=1 defined on [0,𝑇]_𝜏={𝑡_𝑘=𝑘𝜏,0≤𝑘≤𝑁,𝑁ℎ=𝑇} with values in 𝐸 equipped with the norm 𝜙^𝜏_{𝐶𝜏(𝐸)}=max_{1≤𝑘≤𝑁}𝜙𝑡_𝑘𝐸. (2.52) Then, the following theorem on well-posedness of problem (2.48) is established.

Theorem 2.4. For the solution of problem (2.48), the following coercive stability estimates 𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1𝜏^𝑁_{𝑘=1𝐶𝜏(}^∘_𝐶ℎ^2𝛼₎+𝐷²_ℎ𝑢^ℎ_𝑘^𝑁_{𝑘=1𝐶𝜏(∘𝐶ℎ2𝛼)}𝜌𝑡≤𝑀𝑞,𝑠^𝑘−𝜌𝑡_𝑘−1𝜏^𝑁_{𝑘=1𝐶[0,𝑇]𝜏}+𝑀𝑎,𝜙,𝛼,𝑇𝐷²_ℎ𝜑^ℎ_∘𝐶ℎ^2𝛼+𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(∘𝐶ℎ2𝛼)}+𝜌^𝜏_{𝐶[0,𝑇]𝜏},𝑝^𝜏_{𝐶[0,𝑇]𝜏}𝜌𝑡≤𝑀𝑞,𝑠^𝑘−𝜌𝑡_𝑘−1𝜏^𝑁_{𝑘=1𝐶[0,𝑇]𝜏}+𝑀𝑎,𝜙,𝛼,𝑇𝐷²_ℎ𝜑^ℎ_∘𝐶ℎ^2𝛼+𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(∘𝐶ℎ2𝛼)}+𝜌^𝜏_{𝐶[0,𝑇]𝜏} (2.53) hold. Here, 𝐹^ℎ₁𝑡_𝑘=𝑓𝑡_𝑘,𝑥_𝑛0𝑏𝑡_𝑘,𝑥_𝑛𝜌𝑡_𝑘𝑞_𝑠𝑞_𝑛+𝑔𝑡_𝑘,𝑥_𝑛^𝑀−1_𝑛=1,𝜑^ℎ=𝜑𝑥_𝑛^𝑀−1_𝑛=1,𝜌^𝜏=𝜌𝑡_𝑘^𝑁_𝑘=0,𝐷²_ℎ𝑢^ℎ=𝑢_𝑛+1−2𝑢_𝑛+𝑢_𝑛−1ℎ^2𝑀−1_𝑛=11𝑎=𝑞_𝑠𝑎𝐷²_ℎ𝑞^ℎ−𝜎𝑞^ℎ. (2.54)

Proof. The solution of problem (2.48) is searched in the following form: 𝑢^𝑘_𝑛=𝜂^𝑘𝑞_𝑛+𝑤^𝑘_𝑛, (2.55) where 𝜂^𝑘=^𝑘_𝑖=1𝑝^𝑖𝜏,1≤𝑘≤𝑁,𝜂⁰=0. (2.56) Difference derivatives of (2.55) can be written as 𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏=𝜂^𝑘−𝜂^𝑘−1𝜏𝑞_𝑛+𝑤^𝑘_𝑛−𝑤_𝑛^𝑘−1𝜏=𝑝^𝑘𝑞_𝑛+𝑤^𝑘_𝑛−𝑤_𝑛^𝑘−1𝜏,𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²=𝜂^𝑘𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²+𝑤^𝑘_𝑛+1−2𝑤^𝑘_𝑛+𝑤^𝑘_𝑛−1ℎ², (2.57) for any 𝑛,1≤𝑛≤𝑀−1. At the interior grid point 𝑠=⌊𝑥^∗/ℎ⌋, we have that 𝑢^𝑘_𝑠=𝜂^𝑘𝑞_𝑠+𝑤^𝑘_𝑠=𝜌𝑡_𝑘,𝜂^𝑘=𝜌𝑡_𝑘−𝑤^𝑘_𝑠𝑞_𝑠. (2.58) Taking the difference derivative of the last equality and using the triangle inequality, we obtain 𝑝^𝑘=1𝑞_𝑠𝜌𝑡_𝑘−𝜌𝑡_𝑘−1𝜏−𝑤^𝑘_𝑠−𝑤_𝑠^𝑘−1𝜏𝑝_𝑘,𝜌𝑡≤𝑀𝑞,𝑠^𝑘−𝜌𝑡_𝑘−1𝜏+𝑤^𝑘_𝑠−𝑤_𝑠^𝑘−1𝜏≤𝑀𝑞,𝑠max_{1≤𝑘≤𝑁}𝜌𝑡_𝑘−𝜌𝑡_𝑘−1𝜏+max_{1≤𝑘≤𝑁}max_{0≤𝑠≤𝑀}𝑤^𝑘_𝑠−𝑤_𝑠^𝑘−1𝜏≤𝑀𝑞,𝑠max_{1≤𝑘≤𝑁}𝜌𝑡_𝑘−𝜌𝑡_𝑘−1𝜏+max_{1≤𝑘≤𝑁}𝑤^ℎ_𝑘−𝑤^ℎ_𝑘−1𝜏^∘_𝐶ℎ^2𝛼, (2.59)(2.60) for any 𝑘,1≤𝑘≤𝑁.
In estimate (2.60), {𝑤^ℎ_𝑘}^𝑁_𝑘=0 is the solution of the following difference scheme: 𝑤^𝑘_𝑛−𝑤_𝑛^𝑘−1𝜏=𝑎𝑥_𝑛𝑤^𝑘_𝑛+1−2𝑤^𝑘_𝑛+𝑤^𝑘_𝑛−1ℎ²+𝑎𝑥_𝑛𝜌𝑡_𝑘−𝑤^𝑘_𝑠𝑞_𝑠𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²+𝑓𝑡_𝑘,𝑥_𝑛,1≤𝑘≤𝑁,1≤𝑛≤𝑀_𝑙−1,𝑀_𝑙𝑤ℎ=𝑙,𝑁𝜏=𝑇,_𝑘𝑛−𝑤_𝑛^𝑘−1𝜏=𝑎𝑥_𝑛𝑤^𝑘_𝑛+1−2𝑤^𝑘_𝑛+𝑤^𝑘_𝑛−1ℎ²+𝑎𝑥_𝑛𝜌𝑡_𝑘−𝑤^𝑘_𝑠𝑞_𝑠𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝑏𝑡_𝑘,𝑥_𝑛𝑤^𝑘_𝑛+𝑏𝑡_𝑘,𝑥_𝑛𝜌𝑡_𝑘−𝑤^𝑘_𝑠𝑞_𝑠𝑞_𝑛+𝑔𝑡_𝑘,𝑥_𝑛,1≤𝑘≤𝑁,𝑀_𝑙𝑤+1≤𝑛≤𝑀−1,𝑀ℎ=𝐿,𝑁𝜏=𝑇,_0𝑛=𝜑𝑥_𝑛𝑤,0≤𝑛≤𝑀,_𝑘1−𝑤^𝑘₀=𝑤^𝑘_𝑀𝑤=0,0≤𝑘≤𝑁,_{𝑘𝑀𝑙+1}−𝑤^𝑘_𝑀𝑙=𝑤^𝑘_𝑀𝑙−𝑤^𝑘_𝑀𝑙−1,0≤𝑘≤𝑁, (2.61) where 𝑥_𝑛=𝑛ℎ,𝑡_𝑘=𝑘𝜏. Therefore, estimate (2.60) and the following theorem finish the proof of Theorem 2.5.

Theorem 2.5. For the solution of problem (2.61), the following coercive stability estimate 𝑤^ℎ_𝑘−𝑤^ℎ_𝑘−1𝜏^𝑁_{𝑘=1𝐶𝜏(}^∘_𝐶ℎ^2𝛼₎×𝜑≤𝑀𝑎,𝜙,𝛼,𝑇_{ℎ∘𝐶ℎ2𝛼}+𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(∘𝐶ℎ2𝛼)}+𝜌^𝜏_{𝐶[0,𝑇]𝜏} (2.62) holds.

Proof. We can rewrite difference scheme (2.61) in the abstract form: 𝑤^ℎ_𝑘−𝑤^ℎ_𝑘−1𝜏+𝐴^𝑥_ℎ𝑤^ℎ_𝑘+𝐵^𝑥_ℎ𝑤^ℎ_𝑘=𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡−𝜎𝑞^𝑘−𝑤^𝑘_𝑠𝑞_𝑠+𝐹^ℎ₁𝑡_𝑘+𝐹^ℎ₂𝑡_𝑘,𝑡_𝑘𝑤=𝑘𝜏,1≤𝑘≤𝑁,𝑁𝜏=𝑇,_ℎ0=𝜑^ℎ, (2.63) in a Banach space 𝐸=𝐶^∘[0,𝑙]_ℎ with the positive operator 𝐴^𝑥_ℎ defined by 𝐴^𝑥_ℎ𝑢^ℎ=−𝑎(𝑥_𝑛)𝑢_𝑛+1−2𝑢_𝑛+𝑢_𝑛−1ℎ²+𝜎𝑢^𝑀−1_𝑛=1, (2.64) acting on grid functions 𝑢^ℎ such that it satisfies the condition 𝑢₁−𝑢₀=𝑢_𝑀=0. (2.65) For every fixed 𝑡∈[0,𝑇], the difference operators 𝐵^𝑥_ℎ(𝑡) are given by the formula 𝐵^𝑥_ℎ(𝑡)𝑢^ℎ=−𝜎𝑢^𝑛,1≤𝑛≤𝑀_𝑙,−𝜎−𝑏^𝑛(𝑡)𝑢^𝑛,𝑏^𝑛(𝑡)=𝑏𝑡,𝑥_𝑛,𝑥_𝑛=𝑛ℎ,𝑀_𝑙+1≤𝑛≤𝑀−1,^𝑀−1_𝑛=1, (2.66) where 𝜎 is a positive constant and the right-hand side functions are 𝐹^ℎ₁𝑡_𝑘=𝑓𝑡_𝑘,1≤𝑛≤𝑀_𝑙,𝑏𝑡_𝑘𝑞_𝑛𝜌𝑡_𝑘𝑞_𝑠+𝑔𝑡_𝑘,𝑀_𝑙+1≤𝑛≤𝑀−1,^𝑀−1_𝑛=1,𝐹^ℎ₂𝑡_𝑘=0,1≤𝑛≤𝑀_𝑙,−𝑏𝑡_𝑘𝑞_𝑛𝑤^𝑘_𝑠𝑞_𝑠,𝑀_𝑙+1≤𝑛≤𝑀−1,^𝑀−1_𝑛=1. (2.67) Let us denote 𝑅=(𝐼+𝜏𝐴^𝑥_ℎ)⁻¹. In problem (2.63), we have that 𝑤^ℎ_𝑘=𝑅𝑤^ℎ_𝑘−1𝑎𝑞+𝑅𝜏^𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡_𝑘−𝑤^𝑘_𝑠𝑞_𝑠−𝐵^𝑥_ℎ(𝑡)𝑤^ℎ_𝑘+𝐹^ℎ₁𝑡_𝑘+𝐹^ℎ₂𝑡_𝑘, (2.68) for all 𝑘,1≤𝑘≤𝑁. By recurrence relations, we get 𝑤^ℎ_𝑘=𝑅^𝑘𝜑^ℎ+^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡_𝑚−^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝑤^𝑚_𝑠−^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝜏𝑤^𝑚_𝑠+^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₁𝑡_𝑚+^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₂𝑡_𝑚. (2.69) Then, the following presentation of the solution of problem (2.63) 𝐴^𝑥_ℎ𝑤^ℎ_𝑘=𝐴^𝑥_ℎ𝑅^𝑘𝜑^ℎ+^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡_𝑚−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝑤^𝑚_𝑠−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝜏𝑤^𝑚_𝑠+^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₁𝑡_𝑚+^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₂𝑡_𝑚=⁶_𝑘=1𝐽_𝑘 (2.70) is obtained. Here, 𝐽^𝑘₁=𝐴^𝑥_ℎ𝑅^𝑘𝜑^ℎ,𝐽^𝑘₂=^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡_𝑚,𝐽^𝑘₃=−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝑤^𝑚_𝑠,𝐽^𝑘₄=−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝜏𝑤^𝑚_𝑠,𝐽^𝑘₅=^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₁𝑡_𝑚,𝐽^𝑘₆=^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₂𝑡_𝑚. (2.71)
Now, let us estimate 𝐽^𝑘_𝑟 for 𝑟=1,2,…,6 separately. We start with 𝐽^𝑘₁. Applying the definition of norm of the spaces 𝐸^′_𝛼, we get 𝐽^𝑘_1𝐸′𝛼=𝑅^𝑘𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼≤𝑅^𝑘_{𝐸′𝛼→𝐸′𝛼}𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼≤𝑅^𝑘_𝐸→𝐸𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼. (2.72)
Using estimate 𝑅^𝑘_𝐸→𝐸≤𝑀, (2.73) we get 𝐽^𝑘_1𝐸′𝛼≤𝑀₁𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼, (2.74) for any 𝑘,   1≤𝑘≤𝑁.
Let us estimate 𝐽^𝑘₂∶𝐽^𝑘_2𝐸′𝛼=^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝜌𝑡_{𝑚𝐸′𝛼}≤max_{1≤𝑚≤𝑁}𝜌𝑡_𝑚^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸′𝛼, (2.75) where 𝑞𝑎=𝑎^𝑛+1−2𝑞_𝑛+𝑞_𝑛−1𝑞_𝑠ℎ². (2.76) From the definition of norm of the spaces 𝐸^′_𝛼, it follows that ^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸′𝛼≤^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸+sup^𝑘_𝜆>0𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸. (2.77) Let us estimate each term separately. We divide first term into two parts: ^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸=^𝑘−1_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸+𝐴^𝑥_ℎ𝑅𝜏𝑎_𝐸. (2.78) In the first part, by the definition of norm of the spaces 𝐸^′_𝛼 and the identity (see [29]) (𝐼+𝜏𝐴)^−𝑘=1(𝑘−1)!^∞₀𝑡^𝑘−1𝑒^−𝑡exp{−𝜏𝑡𝐴}𝑑𝑡,𝑘≥2, (2.79) we deduce that ^𝑘−1_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸≤^𝑘−1_𝑚=1𝜏(𝑘−𝑚)!^∞₀𝑡^𝑘−𝑚(𝜏𝑡)^1−𝛼𝑒^−𝑡(𝜏𝑡)^1−𝛼𝐴^𝑥_ℎ𝑒^{−𝜏𝑡𝐴}𝑎_𝐸≤‖‖𝑑𝑡𝑎_𝐸′𝛼^𝑘−1_𝑚=1𝜏(𝑘−𝑚)!^∞₀𝑡^𝑘−𝑚(𝜏𝑡)^1−𝛼𝑒^−𝑡=‖‖𝑑𝑡𝑎_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼(𝑘−𝑚)!^∞₀𝑡^{𝑘−𝑚−1+𝛼}𝑒^−𝑡=‖‖𝑑𝑡𝑎_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼(𝑘−𝑚)!^∞₀𝑡^{(𝑘−𝑚−1)𝛼+𝛼}𝑒^−𝛼𝑡𝑡^{(𝑘−𝑚−1)(1−𝛼)}𝑒^{−(1−𝛼)𝑡}𝑑𝑡. (2.80) The Hölder inequality with 𝑝=1/𝛼,𝑞=1/(1−𝛼) and the definition of the gamma function yield that ^𝑘−1_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸≤‖𝑎‖_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼(𝑘−𝑚)!^∞₀𝑡^{(𝑘−𝑚−1)𝛼+𝛼}𝑒^{−𝛼𝑡1/𝛼}𝑑𝑡^𝛼×^∞₀𝑡^{(𝑘−𝑚−1)(1−𝛼)}𝑒^{−(1−𝛼)𝑡1/(1−𝛼)}𝑑𝑡^1−𝛼=‖𝑎‖_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼(𝑘−𝑚)!^∞₀𝑡^𝑘−𝑚𝑒^−𝑡𝑑𝑡^𝛼∞₀𝑡^{𝑘−𝑚−1}𝑒^−𝑡𝑑𝑡^1−𝛼=‖𝑎‖_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼()!(Γ(𝑘−𝑚𝑘−𝑚+1))^𝛼(Γ(𝑘−𝑚))^1−𝛼. (2.81) By the fact that Γ(𝑛)=(𝑛−1)! and Γ(𝑛)=(𝑛−1)Γ(𝑛−1), we get ^𝑘−1_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸≤‖𝑎‖_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼()!()𝑘−𝑚𝑘−𝑚^𝛼Γ()=‖‖𝑘−𝑚𝑎_𝐸′𝛼^𝑘−1_𝑚=1𝜏^𝛼(𝑘−𝑚)^1−𝛼=‖𝑎‖_𝐸′𝛼^𝑘−1_𝑚=1𝜏((𝑘−𝑚)𝜏)^1−𝛼≤𝑀₂‖𝑎‖_𝐸′𝛼0^𝑘𝜏1(𝑘𝜏−𝑠)^1−𝛼𝑑𝑠=𝑀₂‖𝑎‖_𝐸′𝛼−(𝑘𝜏−𝑠)^𝛼𝛼₀^𝑘𝜏. (2.82) So, we have that ^𝑘−1_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸≤𝑀₂‖𝑎‖_𝐸′𝛼(𝑘𝜏)^𝛼𝛼≤𝑀₃(𝛼,𝑇)‖𝑎‖_𝐸′𝛼. (2.83) In the second part, we have that 𝐴^𝑥_ℎ𝑅𝜏𝑎_𝐸≤𝐴^𝑥_ℎ𝑅𝜏_𝐸→𝐸‖𝑎‖_𝐸≤𝑀₄‖𝑎‖_𝐸′𝛼. (2.84) Combining estimates (2.83) and (2.84), we obtain ^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑎_𝐸≤𝑀₅(𝛼,𝑇)‖𝑎‖_𝐸′𝛼. (2.85) Let us estimate the second term. From the Cauchy-Riesz formula (see [29]) 𝑓(𝐴)=12𝜋𝑖_Γ𝑓(𝑧)(𝑧−𝐴)⁻¹𝑑𝑧, (2.86) it follows that ^𝑘_𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝑅^{𝑘−𝑚+1}𝐴^𝑥_ℎ=1𝜏𝑎2𝜋𝑖_{𝑆1∪𝑆2}^𝑘_𝑚=1𝑧(1+𝑧)^{𝑘−𝑚+1}𝜆^𝛼𝜆+𝑧𝜏⁻¹𝐴^𝑥_ℎ𝑧−𝜏𝐴^𝑥_ℎ⁻¹=1𝑎𝑑𝑧2𝜋𝑖_{𝑆1∪𝑆2}^𝑘_𝑚=1(𝑧/𝜏)^−𝛼(1+𝑧)^{𝑘−𝑚+1}𝜆^𝛼𝑧𝜆𝜏+𝑧𝜏^𝛼𝐴^𝑥_ℎ𝑧𝜏−𝐴^𝑥_ℎ⁻¹𝑎𝑑𝑧. (2.87) Since 𝑧=𝜌𝑒^±𝑖𝜙, with |𝜙|≤𝜋/2, the estimate (see [29]) (𝜆−𝐴)⁻¹_𝐸→𝐸≤𝑀𝜙1+|𝜆| (2.88) yields 𝑧𝜏^𝛼𝐴^𝑥_ℎ𝑧𝜏−𝐴^𝑥_ℎ⁻¹𝑎_𝐸≤𝑀₆𝜌𝜏^𝛼𝐴^𝑥_ℎ𝜌𝜏+𝐴^𝑥_ℎ⁻¹𝑎_𝐸,1||≤𝑀𝜆𝜏+𝑧⁶.𝜆𝜏+𝜌 (2.89) Hence, ^𝑘_𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝑅^{𝑘−𝑚+1}𝐴^𝑥_ℎ𝜏𝑎_𝐸≤𝑀₆^∞₀^𝑘_𝑚=1𝜌^1−𝛼1+2𝜌cos𝜙+𝜌^{2(𝑘−𝑚+1)/2}(𝜆𝜏)^𝛼𝑑𝜌‖‖𝜆𝜏+𝜌𝑎_𝐸′𝛼. (2.90) Summing the geometric progression, we get ^𝑘_𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝑅^{𝑘−𝑚+1}𝐴^𝑥_ℎ𝜏𝑎_𝐸≤𝑀₆^∞₀^𝑘_𝑚=1𝜌^1−𝛼1+2𝜌cos𝜙+𝜌^21/2×11−1+2𝜌cos𝜙+𝜌^21/2−1(𝜆𝜏)^𝛼𝑑𝜌‖‖𝜆𝜏+𝜌𝑎_𝐸′𝛼≤𝑀₆^∞₀(𝜆𝜏)^𝛼𝜘𝜌𝑑𝜌𝜆𝜏+𝜌𝜌^𝛼‖𝑎‖_𝐸′𝛼. (2.91) Since the function 𝜘𝜌=𝜌1+2𝜌cos𝜙+𝜌^21/2=−11+1+2𝜌cos𝜙+𝜌^21/22cos𝜙+𝜌 (2.92) does not increase for 𝜌≥0, we have 𝜘(0)=1/cos𝜙≥𝜘(𝜌) for all 𝜌>0. Consequently, ^𝑘_𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝑅^{𝑘−𝑚+1}𝐴^𝑥_ℎ𝜏𝑎_𝐸≤𝑀₆cos𝜙^∞₀(𝜆𝜏)^𝛼𝑑𝜌𝜆𝜏+𝜌𝜌^𝛼‖𝑎‖_𝐸′𝛼 (2.93) for any 𝜆>0. Hence, sup_𝜆>0^𝑘_𝑚=1𝜆^𝛼𝐴^𝑥_ℎ𝜆+𝐴^𝑥_ℎ⁻¹𝑅^{𝑘−𝑚+1}𝐴^𝑥_ℎ𝜏𝑎_𝐸≤𝑀₇𝜙,𝛼‖𝑎‖_𝐸′𝛼. (2.94) Then, using estimates (2.85) and (2.94), we get ^𝑘_𝑚=1𝑅^{𝑘−𝑚+1}−𝑅^𝑘−𝑚𝑎_𝐸′𝛼≤𝑀₈𝜙,𝛼,𝑇‖𝑎‖_𝐸′𝛼,𝐽^𝑘_2𝐸′𝛼≤max_{1≤𝑚≤𝑁}𝜌𝑡_𝑚𝑀₈𝜙,𝛼,𝑇‖𝑎‖_𝐸′𝛼. (2.95)(2.96) Now, let us estimate 𝐽^𝑘₃∶𝐽^𝑘_3𝐸′𝛼=−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²𝑤^𝑚_{𝑠𝐸′𝛼}≤^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝑞_𝑠𝑎𝑞_𝑛+1−2𝑞_𝑛+𝑞_𝑛−1ℎ²−𝜎𝑞_𝐸′𝛼𝑤^𝑚_𝑠. (2.97) Since 𝑤^𝑚_𝑠≤max_{0≤𝑠≤𝑀}𝑤^𝑚_𝑠=𝑤^ℎ_𝐸≤𝑤^ℎ_𝐸′𝛼≤𝐴^𝑥_ℎ⁻¹_{𝐸′𝛼→𝐸′𝛼}𝐴^𝑥_ℎ𝑤^ℎ_𝐸′𝛼≤𝑀𝐴^𝑥_ℎ𝑤^ℎ_𝐸′𝛼, (2.98) and using estimate (2.95), we obtain 𝐽_3𝐸′𝛼≤𝑀₉𝜙,𝛼,𝑇,𝜏‖𝑎‖_𝐸′𝛼^𝑘_𝑚=1𝐴^𝑥_ℎ𝑤^ℎ_𝐸′𝛼𝜏. (2.99)𝐽^𝑘₄ can be estimated as follows: 𝐽^𝑘_4𝐸′𝛼=−^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝜏𝑤^𝑚_{𝑠𝐸′𝛼}=^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝐴^𝑥_ℎ⁻¹𝐴^𝑥_ℎ𝜏𝑤^𝑚_{𝑠𝐸′𝛼}≤^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝐵^𝑥_ℎ(𝑡)𝐴^𝑥_ℎ⁻¹_{𝐸′𝛼→𝐸′𝛼}𝐴^𝑥_ℎ𝑤^𝑚_{𝑠𝐸′𝛼}𝜏. (2.100) From 𝑅^𝑘_{𝐸′𝛼→𝐸′𝛼}≤𝑅^𝑘_𝐸→𝐸𝐴≤𝑀,_𝑥ℎ𝐵^𝑥_ℎ(𝑡)𝐴^𝑥_ℎ⁻¹_{𝐸′𝛼→𝐸′𝛼}≤𝑀 (2.101) it follows that 𝐽^𝑘_4𝐸′𝛼≤𝑀^𝑘_10𝑚=1𝐴^𝑥_ℎ𝑤^𝑚_{𝑠𝐸′𝛼}𝜏. (2.102) The estimations of 𝐽^𝑘₅ and 𝐽^𝑘₆ are as follows. By the definition of the norm of the spaces 𝐸^′_𝛼 and (2.95), we get 𝐽^𝑘_5𝐸′𝛼≤^𝑘_𝑚=1𝐴^𝑥_ℎ𝑅^{𝑘−𝑚+1}𝜏𝐹^ℎ₁(𝑡_𝑚)_𝐸′𝛼+𝑀₁₁𝜙,𝛼,𝑇𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)},𝐽_6𝐸′𝛼≤𝑀₁₂𝜙,𝛼^𝑘_𝑚=1𝐴^𝑥_ℎ𝑤^ℎ_𝐸′𝛼𝜏. (2.103)(2.104) Combining estimates (2.74), (2.96), (2.99), and (2.102)–(2.104), we get 𝐴^𝑥_ℎ𝑤^ℎ_{𝑘𝐸′𝛼}≤𝑀₁𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼+max_{1≤𝑚≤𝑁}𝜌𝑡_𝑚𝑀₈𝜙,𝛼,𝑇‖𝑎‖_𝐸′𝛼+𝑀₉𝜙,𝛼,𝑇,𝜏‖𝑎‖_𝐸′𝛼+𝑀₁₀+𝑀₁₂𝜙,𝛼^𝑘_𝑚=1𝐴^𝑥_ℎ𝑤^ℎ_𝑘𝐸𝛼𝜏+𝑀₁₁𝜙,𝛼,𝑇𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)}. (2.105) Using the discrete analogue of Gronwall's inequality, we get 𝐴^𝑥_ℎ𝑤^ℎ_{𝑘𝐸′𝛼}≤𝑒^𝑀13(𝑎_{,𝜙,𝛼,𝑇,𝜏)}×𝑀₁𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼+𝑀₁₄𝑎,𝜙,𝛼,𝑇𝜌^𝜏_{𝐶[0,𝑇]𝜏}+𝑀₁₁𝜙,𝛼,𝑇𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)}. (2.106) It follows from (2.63) and the triangle inequality that 𝑤^ℎ_𝑘−𝑤^ℎ_𝑘−1𝜏_𝐸′𝛼≤𝑒^𝑀12(𝑎_{,𝜙,𝛼,𝑇)}×𝑀₁𝐴^𝑥_ℎ𝜑^ℎ_𝐸′𝛼+𝑀₁₃𝑎,𝜙,𝛼,𝑇𝜌^𝜏_{𝐶[0,𝑇]𝜏}+𝑀₁₁𝜙,𝛼,𝑇𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)}, (2.107) for every 𝑘,   1≤𝑘≤𝑁. Then, we have that 𝑤^ℎ_𝑘−𝑤^ℎ_𝑘−1𝜏^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)}≤𝑀₁₄×𝐴𝑎,𝜙,𝛼,𝑇_𝑥ℎ𝜑^ℎ_𝐸′𝛼+𝐹^ℎ₁𝑡_𝑘^𝑁_{𝑘=1𝐶𝜏(𝐸′𝛼)}+𝜌^𝜏_{𝐶[0,𝑇]𝜏}. (2.108)
The following theorem finishes the proof of Theorem 2.6.

Theorem 2.6 (see [30]). For 0<𝛼<1/2, the spaces 𝐸^′_𝛼(𝐶[0,𝐿]_ℎ,𝐴^𝑥_ℎ) and 𝐶^2𝛼[0,𝐿]_ℎ coincide and their norms are equivalent.

3. Conclusion

Since artery disease caused by atherosclerosis is one of the most important causes of the death in the world, investigation of the effect of flow over the glycocalyx takes an important place. The flow equations can be formulated as an inverse problem. Here, our aim is to give more detailed understanding of the flow phenomena. Therefore, the well-posedness of the inverse problem of reconstructing the right side of a parabolic equation was investigated. Further, a new computer code regarding the flow analysis for the unknown pressure difference will be written.

Acknowledgment

The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for his comments and suggestions to improve the quality of the paper.