Existence and Uniqueness of the Solutions for Some Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition

Abstract

In this paper, we analyze some initial-boundary value problems for the subdiffusion equation with a fractional dynamic boundary condition in a one-dimensional bounded domain. First, we establish the unique solvability in the Hölder space of the initial-boundary value problems for the equation 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)=𝐿𝑢(𝑥,𝑡)+𝑓₀(𝑥,𝑡), 𝛼∈(0,1), where L is a uniformly elliptic operator with smooth coefficients with the fractional dynamic boundary condition. Second, we apply the contraction theorem to prove the existence and uniqueness locally in time in the Hölder classes of the solution to the corresponding nonlinear problems.

1. Introduction

Let 𝑎 and 𝑏 be any numbers from 𝑅¹ and let Ω=(𝑎,𝑏), Ω_𝑇=Ω×(0,𝑇); Γ_1𝑇={𝑎}×[0,𝑇]; Γ_2𝑇={𝑏}×[0,𝑇]; 𝑇>0 be a fixed value. In this paper, we consider a partial differential equation with the fractional derivative in time 𝑡 as follows: 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)−𝑎₀()𝜕𝑥,𝑡²𝑢(𝑥,𝑡)𝜕𝑥²+𝑎₁()()𝑥,𝑡𝜕𝑢𝑥,𝑡𝜕𝑥+𝑎₂(𝑥,𝑡)𝑢(𝑥,𝑡)=𝑓₀(),()𝑥,𝑡𝑥,𝑡∈Ω_𝑇,𝛼∈(0,1]. (1) Here, 𝜕^𝛼_𝑡 denotes the Caputo fractional derivative with respect to 𝑡 and is defined by (see, e.g., (2.4.6) in [1]), 𝜕^𝛼_𝑡1𝑔=Γ()𝑑1−𝛼𝑑𝑡^𝑡₀𝑔(𝜏)𝑑𝜏(𝑡−𝜏)^𝛼−𝑔(0)Γ(1−𝛼)𝑡^𝛼,𝛼∈(0,1), (2) where Γ is the gamma function, 𝑎_𝑖(𝑥,𝑡), 𝑖=0,2, are the given functions, and 𝑎₀(𝑥,𝑡) is a positive. Note that if 𝛼=1, then (1) represents a parabolic equation. As we are interested in the fractional cases, we restrict the order 𝛼 to the case 0<𝛼<1.

We will solve (1) satisfying the following conditions: 𝑢(𝑥,0)=𝑢₀(𝑥)inΩ(initialcondition), (3) the fractional dynamic boundary condition on Γ_2𝑇: 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)+𝑏₀(𝑡)𝜕𝑢(𝑥,𝑡)𝜕𝑥+𝑏₁(𝑡)𝑢(𝑥,𝑡)=𝑓₁(𝑡)onΓ_2𝑇 (4) and one of the following conditions on Γ_1𝑇: the Dirichlet boundary condition: 𝑢(𝑥,𝑡)=𝑓₂(𝑡)onΓ_1𝑇, (5) or the Neumann boundary condition: 𝜕𝑢(𝑥,𝑡)𝜕𝑥=𝑓₃(𝑡)onΓ_1𝑇, (6) or the fractional dynamic boundary condition: 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)−𝑏₂(𝑡)𝜕𝑢(𝑥,𝑡)𝜕𝑥+𝑏₃(𝑡)𝑢(𝑥,𝑡)=𝑓₄(𝑡)onΓ_1𝑇. (7)

Here, 𝑏₀(𝑡) and 𝑏₂(𝑡) are given positive functions, and 𝑏₁(𝑡), 𝑏₃(𝑡), and 𝑓_𝑖(𝑡), 𝑖=2,4, are given functions.

Concerning problems (1)–(6), they have the following features. First of all, (4) is a fractional dynamic boundary condition; next, these problems are formulated for the subdiffusion linear equation.

Note that if 𝛼=1, conditions (4) and (6) are called normal dynamic boundary conditions. These conditions are very natural in many mathematical models, including heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, and problems in fluid dynamics, and in the Stefan problem, (see [2–4] and the references therein). At the present moment, there are a lot of works concerning linear and nonlinear problems with dynamic boundary conditions. Here we make no pretence to provide a complete survey on the results related to problems of the type (1)-(6), if 𝛼=1, and present only some of them. The initial-boundary value problems for the heat equation in the certain shape of domains with linear dynamic boundary condition have been solved with the separation variables method or with the Laplace transformation in [3]. In the case of smooth domains, these problems have been researched with the approaches of the general theory for evolution equations in Hilbert and Banach spaces, and the weak solutions of the above mentioned problems have been obtained in [5–7]. Using the Schauder method, Grigor’eva and Mogilevskii [8] have got the coercive estimates of the solution in the anisotropic Sobolev spaces. The one-to-one solvability in the case of the linear parabolic equation with variable coefficients has been proved by Bazaliy [4] in the Hölder spaces and by Bizhanova and Solonnikov [9] in the weighted Hölder classes. The global and local existence for the solution to initial-boundary value problem for linear and quasilinear equations with nonlinear dynamic boundary conditions has been discussed in [10–12] (see also references there).

Over the past few decades, an intensive effort has been put into developing theoretical models for systems with diffusive motion that cannot be modelled as the standard Brownian motion [13, 14]. The signature of this anomalous diffusion is that the mean square displacement of the diffusing species ⟨(Δ𝐱)²⟩ scales as a nonlinear power law in time, that is, ⟨(Δ𝐱)²⟩∼𝑡^𝛼. If 𝛼∈(0,1), this is referred to as subdiffusion. In recent years, the additional motivation for these studies has been stimulated by experimental measurements of subdiffusion in porous media [15], glass forming materials [16], and biological media [17]. The review paper by Klafter et al. [18] provides numerous references to physical phenomena in which anomalous diffusion occurs.

Here we refer to several works on the mathematical treatments for linear equation (1). Kochubei [19, 20], and Pskhu [21, 22] constructed the fundamental solution in 𝑅^𝑑 and proved the maximum principle for the Cauchy problem. Gejji and Jafari [23] solved a nonhomogeneous fractional diffusion-wave equation in a one-dimensional bounded domain. Metzler and Klafter [14], using the method of images and the Fourier-Laplace transformation technique, obtained the solutions of different boundary value problems for the homogenous fractional diffusion equation in a half-space and in a box. Agrawal [24] constructed a solution of a fractional diffusion equation using a finite transform technique and presented numerical results in a one-dimensional bounded domain. Mophou and N’Guérékata [25] and Sakamoto and Yamamoto [26] proved the one-valued solvability of the initial-boundary value problem for the fractional diffusion equation with variable coefficients which is 𝑡-independent with the homogenous Dirichlet conditions in the Sobolev space. Note that, in [26], the authors obtained the certain regularities of the solution given by the eigenfunction expansions and established several results of uniqueness for related inverse problems.

As source books related with fractional derivatives, see the work of Samko et al. [27] which is an encyclopedic treatment of the fractional calculus and also Kilbas et al. [1], Mainardi [28], Podlubny [29], and Pskhu [21].

As for the quasilinear equation like (1), Clément et al. [30] analyzed the abstract fractional parabolic quasilinear equations. Via maximal regularity results in the corresponding linear equation, they arrived to results on existence (locally in time), uniqueness, and continuation on the quasilinear equation in the BUC classes with a weight. As for investigation of the problem with fractional dynamic boundary conditions, Kirane and Tatar [31] have analyzed the issue of nonexistence of local and global solutions for elliptic systems with nonlinear fractional dynamic boundary conditions.

To the authors’ best knowledge, there are no works published concerning the solvability of problems (1)–(6) in the Hölder classes. The first purpose of this paper is to prove the well-posedness and the regularity of the solutions to problems (1)–(6) in the smooth classes. Second, we obtain a local in time solvability in the smooth classes of the corresponding nonlinear problems. This paper is organized as follows. In the second section, we state the main results, Theorems 3–5, and define the functional spaces. In Section 3, we establish the one-valued solvability of certain model problems in 𝑅⁺_𝑇. The principal results of this section are given in Theorems 9 and 13. In Section 4, we prove the main results of this paper. To this end, we will combine ideas from [32] with coercive estimates of the solutions to the corresponding model problems (Section 3). In Section 5, we address the corresponding nonlinear problems. We first reduce them to a form 𝔄𝑤=𝔗(𝑤), where 𝔗(𝑤) is a nonlinear function of 𝑤 and 𝔄 is the linear operator derived in Section 4; that is, 𝔄⁻¹𝔗 is the solution of the model problem for data 𝔗. Setting 𝔖=𝔄⁻¹𝔗, we will then prove that the mapping 𝑤→𝔖(𝑤), where 𝔖(𝑤)=𝔄⁻¹𝔗𝑤, is a contraction, so that it has a unique fixed point. The principal results of this section are formulated in Theorem 18 and Remarks 19 and 20. The Appendix contains the proofs of some auxiliary assertions which are applied in Section 3.

2. The Functional Spaces and the Main Results

Let us introduce the functional spaces. Let Ω be a bounded or an unbounded domain in 𝑅¹, 𝑥, 𝑥∈Ω; 𝑡,𝜏∈[0,𝑇]; 𝜃,𝛾∈(0,1). Denote ⟨𝑣⟩^(𝜃)_𝑥,Ω𝑇=sup_{(),(𝑥,𝑡)𝑥,𝑡∈Ω𝑇,𝑥≠𝑥}|𝑣(𝑥,𝑡)−𝑣(𝑥,𝑡)||𝑥−𝑥|^𝜃;⟨𝑣⟩^(𝛾)_𝑡,Ω𝑇=sup_{(),()𝑥,𝑡𝑥,𝜏∈Ω𝑇,𝑡≠𝜏}|𝑣(𝑥,𝑡)−𝑣(𝑥,𝜏)||𝑡−𝜏|^𝛾. (8)

Definition 1. We will say that functions 𝑣(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇) and that 𝑤(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇) if and only if the functions 𝑣(𝑥,𝑡) and 𝜕^𝑘𝑤(𝑥,𝑡)/𝜕𝑥^𝑘, 𝑘=0,2, 𝜕^𝛼_𝑡𝑤(𝑥,𝑡) are continuous and the following norms are finite: ‖𝑣‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇)=sup_Ω𝑇|𝑣(𝑥,𝑡)|+⟨𝑣⟩^(𝜃)_𝑥,Ω𝑇+⟨𝑣⟩^{((𝜃/2)𝛼)}_𝑡,Ω𝑇,‖𝑤‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)=²_𝑘=0sup_Ω𝑇𝜕^𝑘𝑤𝜕𝑥^𝑘+sup_Ω𝑇𝜕^𝛼_𝑡𝑤+𝜕²𝑤𝜕𝑥^2(𝜃)_𝑥,Ω𝑇+𝜕^𝛼_𝑡𝑤^(𝜃)_𝑥,Ω𝑇+²_𝑘=1𝜕^𝑘𝑤𝜕𝑥^{𝑘(((2+𝜃−𝑘)/2)𝛼)}_𝑡,Ω𝑇+𝜕^𝛼_𝑡𝑤^{((𝜃/2)𝛼)}_𝑡,Ω𝑇. (9)(10)

Note that if 𝛼=1, the spaces 𝐶^{𝑙+𝜃,((𝑙+𝜃)/2)𝛼}(Ω_𝑇), 𝑙=0,2, coincide with the ordinary Hölder spaces (see (1.10)–(1.12) in Chapter 1 in [32]). Further, we also use the Banach spaces 𝐶^𝛾𝛼([0,𝑇]) and 𝐶^(1+𝛾)𝛼([0,𝑇]) of the functions 𝑉(𝑡) and 𝑊(𝑡) with the finite norms ‖𝑉‖_𝐶^𝛾𝛼_([0,𝑇])=sup_[]0,𝑇|𝑉(𝑡)|+⟨𝑉⟩_[]^(𝛾𝛼)_𝑡,0,𝑇,‖𝑊‖_𝐶^(1+𝛾)𝛼_([0,𝑇])=sup_[]0,𝑇|𝑊(𝑡)|+sup_[]0,𝑇𝜕^𝛼_𝑡𝑊+𝜕^𝛼_𝑡𝑊_[]^(𝛾𝛼)_𝑡,0,𝑇. (11)(12) In a similar way, we introduce the spaces 𝐶^{(𝑘+𝛾)𝛼}(𝜕Ω_𝑇), 𝑘=0,1.

Definition 2. We will say that functions 𝑣₁(𝑥,𝑡)∈𝐶₀^{𝜃,(𝜃/2)𝛼}(Ω_𝑇) and that 𝑣₂(𝑥,𝑡)∈𝐶₀^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇) if and only if 𝑣₁(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇) and 𝑣₁(𝑥,0)=0 and 𝑣₂(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇) and 𝑣₂(𝑥,0)=𝜕^𝛼_𝑡𝑣₂(𝑥,0)=0.
If 𝑢₁(𝑡)∈𝐶^𝛾𝛼([0,𝑇]),𝑢₁(0)=0(𝑢₂(𝑡)∈𝐶^(1+𝛾)𝛼([0,𝑇]), and 𝑢₂(0)=𝜕^𝛼_𝑡𝑢₂(0)=0), then 𝑢₁(𝑡)∈𝐶₀^𝛾𝛼([0,𝑇])(𝑢₂(𝑡)∈𝐶₀^(1+𝛾)𝛼([0,𝑇])).

We introduce the spaces of 𝐶₀^{(𝑘+𝛾)𝛼}(𝜕Ω_𝑇), 𝑘=0,1, with the same way.

Let 𝜹₁ and 𝜹₂ be some positive numbers. We assume that the given functions 𝑎_𝑖(𝑥,𝑡), 𝑖=0,2, and 𝑏_𝑗(𝑡), 𝑗=0,2, in (1), (4), and (7) are subject to the following conditions: (i)𝑎₀(𝑥,𝑡)≥𝜹₁>0inΩ_𝑇;𝑏₀(𝑡),𝑏₂(𝑡)≥𝜹₂[];()𝑎>0∀𝑡∈0,𝑇ii_𝑖(𝑥,𝑡)∈𝐶^{𝜃,(𝜃𝛼/2)}Ω_𝑇,𝑏_𝑗(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]),𝑖=0,2,𝑗=0,3; (13)(14)(iii)𝑓₁(0)−𝑏₀(0)𝜕𝑢₀(𝑏)𝜕𝑥−𝑏₁(0)𝑢₀(𝑏)=𝑎₀()𝜕𝑏,0²𝑢₀(𝑏)𝜕𝑥²−𝑎₁(𝑏,0)𝜕𝑢₀(𝑏)𝜕𝑥−𝑎₂(𝑏,0)𝑢₀(𝑏)+𝑓₀(𝑏,0); (15) and one of the following: 𝑓₂(0)=𝑢₀(𝑎),𝜕^𝛼_𝑡𝑓₂(0)=𝑎₀()𝜕𝑎,0²𝑢₀(𝑎)𝜕𝑥²−𝑎₁(𝑎,0)𝜕𝑢₀(𝑎)𝜕𝑥−𝑎₂(𝑎,0)𝑢₀(𝑎)+𝑓₀()((5;𝑎,0inthecaseofDirichletcondition)) (16) or 𝑓₃(0)=𝜕𝑢₀(𝑎)((6;𝜕𝑥inthecaseofNeumanncondition)) (17) or 𝑓₄(0)+𝑏₂(0)𝜕𝑢₀(𝑎)𝜕𝑥−𝑏₃(0)𝑢₀(𝑎)=𝑎₀()𝜕𝑎,0²𝑢₀(𝑎)𝜕𝑥²−𝑎₁(𝑎,0)𝜕𝑢₀(𝑎)𝜕𝑥−𝑎₂(𝑎,0)𝑢₀(𝑎)+𝑓₀()(7.𝑎,0inthecaseoffractionaldynamicboundarycondition)) (18)Note that requirements (15)–(18) are called the consistency conditions.

The main results of our paper are the following:

Theorem 3. Let 𝜃,𝛼∈(0,1), and conditions (13)–(16) hold, and 𝑓₀(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇), 𝑢₀(𝑥)∈𝐶^2+𝜃(Ω), 𝑓₁(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]), 𝑓₂(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝑇]) for any positive number 𝑇. Then there exists a unique solution 𝑢(𝑥,𝑡) of problem (1)–(5): 𝑢(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)∈𝐶^{((1+𝜃)/2)𝛼}(Γ_2𝑇), and ‖𝑢‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)+𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}+𝑓_{2𝐶((2+𝜃)/2)𝛼([0,𝑇])}, (19) where a positive constant 𝐶 depends only on the measure of Ω and ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇), 𝑖=0,2, ‖𝑏_𝑗‖_{𝐶((1+𝜃)/2)𝛼([0,𝑇])}, 𝑗=0,1.

Theorem 4. Let 𝜃,𝛼∈(0,1), and conditions (13)–(15), and (17) hold, and let 𝑓₀(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇), 𝑢₀(𝑥)∈𝐶^2+𝜃(Ω), 𝑓_𝑖(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]), 𝑖=1,3 for any positive number 𝑇. Then there exists a unique solution 𝑢(𝑥,𝑡) of problem (1)–(4), (6) as follows: 𝑢(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)∈𝐶^{((1+𝜃)/2)𝛼}(Γ_2𝑇), and ‖𝑢‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)+𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}+𝑓_{3𝐶((1+𝜃)/2)𝛼([0,𝑇])}, (20) where 𝐶 is a positive constant and depends only on ‖𝑏_𝑗‖_{𝐶((1+𝜃)/2)𝛼([0,𝑇])}, 𝑗=0,1, ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇), 𝑖=0,2, and the measure of Ω.

Theorem 5. Let 𝜃,𝛼∈(0,1), and conditions (13)–(15) and (18) hold, and 𝑓₀(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇), 𝑢₀(𝑥)∈𝐶^2+𝜃(Ω), 𝑓_𝑖(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]), and 𝑖=1,4 for any positive number 𝑇. Then there exists a unique solution 𝑢(𝑥,𝑡) of problem (1)–(4), (7) as follows: 𝑢(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)∈𝐶^{((1+𝜃)/2)𝛼}(Γ_𝑖𝑇), 𝑖=1,2, and ‖𝑢‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)+²_𝑖=1𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ𝑖𝑇)}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}+𝑓_{4𝐶((1+𝜃)/2)𝛼([0,𝑇])}, (21) where 𝐶 is a positive constant and depends only on ‖𝑏_𝑗‖_{𝐶((1+𝜃)/2)𝛼([0,𝑇])},𝑗=0,3, ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇),  𝑖=0,2, and the measure of Ω.

3. Model Problems

Let 𝐷⁺=(0,+∞), and 𝐷⁺_𝑇=𝐷⁺×(0,𝑇), and 𝑎₀ and 𝑏₀ be some positive numbers. Here we will discuss the first initial-boundary value problem for the fractional diffusion equation in 𝐷⁺_𝑇 and the initial-boundary value problem with the fractional boundary condition in 𝐷⁺_𝑇.

3.1. The Solvability of the First Initial-Boundary Value Problem for the Subdiffusion Equation

We look for the function 𝑣(𝑥,𝑡) by the following conditions: 𝜕^𝛼_𝑡𝑣(𝑥,𝑡)−𝑎₀𝜕²𝑣(𝑥,𝑡)𝜕𝑥²=𝑔₀(𝑥,𝑡)in𝐷⁺_𝑇();𝑣(),𝛼∈0,1𝑥,0=𝑣₀(𝑥),𝑥∈𝐷⁺;𝑣()[];𝑣()𝑥,𝑡⟶0if𝑥⟶+∞,∀𝑡∈0,𝑇0,𝑡=𝑔₁(𝑡),𝑡∈[0,𝑇], (22)(23)(24) where 𝑔₀(𝑥,𝑡) and 𝑔₁(𝑡),𝑣₀(𝑥) are some given functions.

We assume that the following conditions hold: (i)𝑣₀(0)=𝑔₁(0);𝜕^𝛼_𝑡𝑔₁(0)=𝑎₀𝜕²𝑣₀𝜕𝑥²+𝑔₀();()𝑣𝑥,0ii₀(𝑥),𝑔₀(𝑥,𝑡)≡0,if|𝑥|<𝑅₀∀𝑡∈[0,𝑇], (25)(26) for some positive number 𝑅₀.

First, we will study problem (22)–(24) under restriction 𝑣₀(𝑥),𝑔₀(𝑥,𝑡)≡0. (27) We will search a solution of (22)–(24) under restriction (27) in the class 𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇), if 𝑔₁(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝑇]) for 𝜃∈(0,1).

Note that conditions (25) and (26) together with restriction (27) allow us to apply the Laplace transformation in 𝑡 to the right hand sides of (22)–(24). Indeed, conditions (25)–(27) mean that the right hand sides in (22)–(24) except 𝑔₁(𝑡) equal zero and 𝑔₁(0)=0. Thus, we can extend the right hand sides in (22)–(24) by 0 for 𝑡≤0 and save, for simplicity, the same notation for the extension of the function 𝑔₁(𝑡). Therefore, we can apply, at least formally, the Laplace transformation in 𝑡 to (22)–(24) in the case of (25)–(27) hold.

Denote by 𝑤(𝑥,𝑝) the Laplace transformation of the function 𝑤(𝑥,𝑡); that is, 𝑤𝑥,𝑝=₀^+∞𝑒^−𝑝𝑡𝑤(𝑥,𝑡)𝑑𝑡. (28) The Laplace transformation in (22)–(24) leads to the problem 𝑝^𝛼𝑣𝑥,𝑝−𝑎₀𝜕²𝑣𝑥,𝑝𝜕𝑥²=0,𝑥∈𝐷⁺;𝑣𝑣=𝑔𝑥,𝑝⟶0if𝑥⟶+∞;0,𝑝₁𝑝. (29) Here we used the following formula from [33]: 𝜕^𝛼_𝑡𝑤=𝑝^𝛼𝑤𝑥,𝑝−𝑝^𝛼−1𝑤(𝑥,0). (30) One can easily check that the following function solves the equations in (29): 𝑣𝑥,𝑝=exp−𝑥𝑝^𝛼/2√𝑎₀𝑔₁𝑝. (31) Due to formula (2.30) in [34] and the inverse Laplace transformation, we get the integral representations of 𝑣(𝑥,𝑡) as follows: 𝑣(𝑥,𝑡)=^𝑡₀𝐺(𝑥,𝑡−𝜏)𝑔₁(𝜏)𝑑𝜏, (32) where 𝐺(𝑥,𝑡)=𝑡⁻¹𝑊−𝑥𝑡^−𝛼/2√𝑎₀𝛼;−2,0. (33) Here 𝑊(𝑧;𝛽,𝛾) is the Wright function, which is defined for 𝑧,𝛽,𝛾∈ℂ as (see formula (1.8.1(27)) in v.3 [35]) 𝑊𝑧;𝛽,𝛾=^∞_𝑘=0𝑧^𝑘.𝑘!Γ𝛽𝑘+𝛾 (34) The main properties of the Wright functions are described in Chapters 4.1, v.1 and 18.1 v3 in [35], Chapter 1.11 in [1], Chapter 1.3 in [34], and Chapter 2 in [21, 36].

In Lemma 6, we describe the properties of the kernel 𝐺(𝑥,𝑡) which will be necessary to estimate the function 𝑣(𝑥,𝑡). Its proof is represented in Appendix A.

Lemma 6. Let 𝜅, 𝛿, and 𝜃 be some positive constants, 𝜅=𝜅(𝛼), 𝜃∈(0,1], 𝑥∈𝐷⁺, 𝑡∈[0,𝑇]. Then one has the following. (i)()()Thekernel𝐺𝑥,𝑡ispositivefor𝑥>0.ii₀^+∞𝐺()()()𝑥,𝑡𝑑𝑡=1.iii𝑡𝐺𝑥,𝑡≤𝐶exp−𝜅𝑥𝑡^{−𝛼/21/(1−𝛼/2)};𝜕𝐺(𝑥,𝑡)𝑡𝜕𝑡≤𝐶⁻²1+𝑥𝑡^{−𝛼/24/𝛼};𝜕𝐺(𝑥,𝑡)𝑡𝜕𝑥≤𝐶^{−1−𝛼/2}1+𝑥𝑡^{−𝛼/21+2/𝛼}.(iv)^𝑡₀𝐺(𝑥,𝜏)𝜏^(𝜃/2)𝛼𝑑𝜏≤𝐶𝑡^(𝜃/2)𝛼.(v)_𝛿^+∞𝜕𝐺(𝑥,𝜏)𝜏𝜕𝑥^(𝜃/2)𝛼𝑑𝜏≤𝐶𝛿^{−(1−𝜃)𝛼/2}.(vi)_𝛿^+∞𝜕𝐺(𝑥,𝜏)𝜏𝜕𝜏^𝜃𝛼/2𝑑𝜏≤𝐶𝛿^{−1+𝜃𝛼/2}.(vii)^𝑡₀(𝑡−𝜏)^−𝛼𝐺(𝑥,𝜏)𝑑𝜏≤𝐶𝑡^−𝛼exp−𝜅𝑥𝑡^{−(𝛼/2)1/(1−𝛼/2)}.(viii)𝐷^𝛼_𝑡𝐺(𝑥,𝑡)=𝑎₀𝜕²𝐺(𝑥,𝑡)𝜕𝑥²in𝐷⁺_𝑇, (35)(36)(37)(38)(39)(40)(41)(42) where 𝐷^𝛼_𝑡 is the Riemann-Liouville fractional derivative, and its definition is in (2.1.8) in [1].

Lemma 7. Let 𝜃∈(0,1), and conditions (25)–(27) hold, and 𝑔₁(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝑇]). Then the function 𝑣(𝑥,𝑡) represented by formula (32) satisfies the following inequalities: sup_𝐷+𝑇𝜕^𝛼_𝑡𝑣≤𝐶sup_[]0,𝑇𝜕^𝛼_𝑡𝑔₁+𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇;𝜕^𝛼_𝑡𝑣^{((𝜃/2)𝛼)}_{𝑡,𝐷+𝑇}≤𝐶𝜕^𝛼_𝑡𝑔_{1𝐶(𝜃/2)𝛼([0,𝑇])};𝜕^𝛼_𝑡𝑣^(𝜃)_{𝑥,𝐷+𝑇}≤𝐶𝜕^𝛼_𝑡𝑔_{1𝐶(𝜃/2)𝛼([0,𝑇])}. (43)(44)(45)

Proof. First, we obtain the representation of 𝜕^𝛼_𝑡𝑣. To this end, we need the following properties of the fractional derivative (see Lemma 2.10 and formula (2.4.10) in [1]): (i)         𝜕^𝛼_𝑡𝜌(𝑡)=𝐷^𝛼_𝑡𝜌(𝑡),if𝜌(0)=0, (46) where (see (2.1.8) in [1]) 𝐷^𝛼_𝑡𝜌(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀𝜌(𝜏)(𝑡−𝜏)^𝛼𝑑𝜏,𝛼∈(0,1). (47)(ii) If the functions 𝜌_𝑖(𝑡), 𝑖=1,2, and 𝐷^𝛼_𝑡𝜌₁(𝑡) are bounded in [0,𝑇], then 𝐷^𝛼_𝑡^𝑡₀𝜌₁(𝑡−𝜏)𝜌₂(𝜏)=𝑑𝜏_𝑡0𝜌₂(𝑡−𝜏)𝐷^𝛼_𝜏𝜌₁(𝜏)𝑑𝜏+𝜌₂(𝑡)lim_𝑧→0𝐼_𝑧^1−𝛼𝜌₁(𝑧),∀𝑡∈[0,𝑇], (48) where 𝐼^𝜈_𝑡 is the Riemann-Liouville fractional integral of order 𝜈 (see, e.g. (2.1.1) in [1]) 𝐼^𝜈_𝑡𝜌(𝑡)=1Γ(𝜈)^𝑡₀𝜌(𝜏)(𝑡−𝜏)^1−𝜈𝑑𝜏,Re𝜈>0. (49)One can easily see that 𝜕^𝛼_𝑡𝑔₁(𝑡)∈𝐶^(𝜃/2)𝛼([0,𝑇]) and 𝜕^𝛼_𝑡𝑔₁(0)=0 (see (25) and (27)). Then, using properties (46) and (48) and equality (36), we represent the function 𝜕^𝛼_𝑡𝑣(𝑥,𝑡) as 𝜕^𝛼_𝑡𝑣()=𝑥,𝑡_𝑡0𝐺(𝑥,𝑡−𝜏)𝜕^𝛼_𝑡𝑔₁(𝜏)−𝜕^𝛼_𝑡𝑔₁(𝑡)𝑑𝜏+𝜕^𝛼_𝑡𝑔₁(𝑡). (50)
Namely, this representation will be useful below. To prove inequality (43), we will use this representation, statement (i) from Lemma 6 and the first estimate in (37) as follows: 𝜕^𝛼_𝑡𝑣≤𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇×^𝑡₀𝜏^{(𝜃/2)𝛼−1}exp−𝜅𝑥𝜏^{−𝛼/2(1/(1−𝛼/2))}𝑑𝜏+max_[0,𝑇]𝜕^𝛼_𝑡𝑔₁≤𝐶𝜕^𝛼_𝑡𝑔_{1𝐶(𝜃/2)𝛼([0,𝑇])}. (51) In view of 𝜕^𝛼_𝑡𝑔₁(𝑡)=0, if 𝑡≤0, we can rewrite representation (50) as 𝜕^𝛼_𝑡𝑣()=𝑥,𝑡_𝑡−∞𝐺(𝑥,𝑡−𝜏)𝜕^𝛼_𝑡𝑔₁(𝜏)−𝜕^𝛼_𝑡𝑔₁(𝑡)𝑑𝜏+𝜕^𝛼_𝑡𝑔₁(𝑡). (52)
Let 𝑥₁, 𝑥₂∈𝐷⁺ and 𝑥₂>𝑥₁. Denote Δ𝑥∶=𝑥₂−𝑥₁,Δ_𝑥𝜕^𝛼_𝑡𝑣∶=𝜕^𝛼_𝑡𝑣𝑥₂,𝑡−𝜕^𝛼_𝑡𝑣𝑥₁,𝑡. (53) Then, Δ_𝑥𝜕^𝛼_𝑡𝑣=^{(Δ𝑥)2/𝛼}₀𝐺𝑥₂,𝜏𝜕^𝛼_𝜏𝑔₁(𝑡−𝜏)−𝜕^𝛼_𝜏𝑔₁(𝑡+)𝑑𝜏_{(Δ𝑥)2/𝛼0}𝐺𝑥₁,𝜏𝜕^𝛼_𝜏𝑔₁(𝑡)−𝜕^𝛼_𝜏𝑔₁(𝑡−𝜏)𝑑𝜏+₍₎^+∞_Δ𝑥2/𝛼𝐺𝑥₂,𝜏−𝐺𝑥₁×𝜕,𝜏_𝛼𝜏𝑔₁(𝑡−𝜏)−𝜕^𝛼_𝜏𝑔₁(𝑡)𝑑𝜏≡³_𝑖=1ℑ_𝑖. (54)
Using inequality (38) with 𝑡∶=(Δ𝑥)^2/𝛼, 𝜃∶=𝜃, we obtain ℑ₁+ℑ₂≤const.𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇𝑥₂−𝑥₁^𝜃. (55) As for the term ℑ₃, we apply the mean-value theorem to the difference [𝐺(𝑥₂,𝜏)−𝐺(𝑥₁,𝜏)] together with inequality (39) (where 𝛿∶=(Δ𝑥)^2/𝛼, 𝜃∶=𝜃) and deduce ℑ₃≤const.𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇(Δ𝑥)^{1+(2/𝛼)(−(𝛼/2)+𝜃(𝛼/2))}. (56) Thus, representation (54) together with inequalities (55) and (56) prove the correctness of (44).
To complete the proof of Lemma 7, we need to obtain inequality (45). Let 𝑡₁,𝑡₂∈[0,𝑇] and 𝑡₂>𝑡₁. Denote Δ𝑡∶=𝑡₂−𝑡₁,Δ_𝑡𝜕^𝛼_𝑡𝑣∶=𝜕^𝛼_𝑡𝑣𝑥,𝑡₂−𝜕^𝛼_𝑡𝑣𝑥,𝑡₁. (57) We analyze the difference Δ_𝑡𝜕^𝛼_𝑡𝑣=^𝑡1_{2𝑡1−𝑡2}𝐺𝑥,𝑡₁−𝜏𝜕^𝛼_𝑡𝑔₁𝑡₁−𝜕^𝛼_𝑡𝑔₁(𝜏+)𝑑𝜏_{𝑡22𝑡1−𝑡2}𝐺𝑥,𝑡₂−𝜏𝜕^𝛼_𝑡𝑔₁(𝜏)−𝜕^𝛼_𝑡𝑔₁𝑡₂+𝑑𝜏_{2𝑡1−𝑡2−∞}𝐺𝑥,𝑡₁−𝜏𝑑𝜏𝜕^𝛼_𝑡𝑔₁𝑡₁−𝜕^𝛼_𝑡𝑔₁𝑡₂+^{2𝑡1−𝑡2}_−∞𝜕^𝛼_𝑡𝑔₁(𝜏)−𝜕^𝛼_𝑡𝑔₁𝑡₂×𝐺𝑥,𝑡₂−𝜏−𝐺𝑥,𝑡₁+𝜕−𝜏𝑑𝜏_𝛼𝑡𝑔₁𝑡₂−𝜕^𝛼_𝑡𝑔₁𝑡₁≡⁵_𝑖=1𝔍_𝑖. (58) As for the last term in this sum, it is estimated by (Δ𝑡)^𝜃𝛼/2⟨𝜕^𝛼_𝑡𝑔₁⟩^(𝜃/2)𝛼_{𝑡,[0,𝑇]}. We change the variable 𝜏=𝑡₁−𝑧 in the term 𝔍₁ and apply estimate (38) with 𝑡∶=Δ𝑡, 𝜃∶=𝜃. Thus, we have 𝔍₁≤const.(Δ𝑡)^𝜃𝛼/2𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇. (59) In the same way, we evaluate the function 𝔍₂. The estimate of the term 𝔍₃ follows from the properties of the function 𝜕^𝛼_𝑡𝑔₁ and inequality (36). At last, the mean-value theorem together with estimate (40), where 𝛿∶=Δ𝑡, 𝜃∶=𝜃, lead to 𝔍₄≤𝐶(Δ𝑡)^𝜃𝛼/2𝜕^𝛼_𝑡𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇. (60) Therefore, inequality (45) is deduced from (58)–(60).

Now, based on the results of Lemmas 6 and 7, we can infer the next assertion.

Lemma 8. Let conditions of Lemma 7 hold. Then there exists a unique solution 𝑣(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇) of problem (22)–(24), which is represented with (32) and ‖𝑣‖_{𝐶2+𝜃,((2+𝜃)/2)𝛼(𝐷}⁺_𝑇)≤𝐶𝑔_{1𝐶((2+𝜃)/2)𝛼([0,𝑇])}. (61)

Proof. First of all we obtain estimate (61). One can get the following inequality using the results of Lemma 7 and (22), where 𝑔₀≡0∶𝜕²𝑣𝜕𝑥²_{𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)≤const.𝜕^𝛼_𝑡𝑣_{𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)≤𝐶𝑔_{1𝐶((2+𝜃)/2)𝛼([0,𝑇])}. (62) Next, we use formula (3.5.4) from [1] as follows: 𝑤(𝑡)(0)=1−𝑤Γ(𝛼)^𝑡₀𝜕^𝛼_𝜏𝑤(𝜏)(𝑡−𝜏)^1−𝛼𝑑𝜏, (63) to evaluate the maximum of |𝑣(𝑥,𝑡)|. Hence, (43) and (63) lead to inequality sup_𝐷+𝑇|𝑣|≤const.sup_𝐷+𝑇𝜕^𝛼_𝑡𝑣≤𝐶𝑔_{1𝐶((2+𝜃)/2)𝛼([0,𝑇])}; (64) here we use the fact that 𝑣(𝑥,0)=0.
After that, the minor seminorms of the function 𝑣(𝑥,𝑡) are estimated with the interpolation inequalities from Section 8.8 [37] and (43)–(45), (62), and (64). Therefore, the arguments above prove inequality (61) and the embedding 𝑣(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇).
Next, we show that the function 𝑣(𝑥,𝑡) given by (32) satisfies (22). To this end, we use equality (46) and (48) and represent 𝜕^𝛼_𝑡𝑣(𝑥,𝑡) as 𝜕^𝛼_𝑡𝑣()=𝑥,𝑡_𝑡0𝑔₁(𝑡−𝜏)𝐷^𝛼_𝑡𝐺(𝑥,𝜏)𝑑𝜏+𝑔₁(𝑡)lim_𝜏→0𝐼_𝜏^1−𝛼𝐺(𝑥,𝜏). (65) Then (41) leads to 𝜕^𝛼_𝑡𝑣(𝑥,𝑡)=^𝑡₀𝑔₁(𝑡−𝜏)𝐷^𝛼_𝑡𝐺(𝑥,𝜏)𝑑𝜏. (66) Next, due to property (42) of the function 𝐺(𝑥,𝑡), one can check that 𝜕^𝛼_𝑡𝑣(𝑥,𝑡)=𝑎₀𝜕²𝑣𝜕𝑥². (67) As it follows from the first inequality in (37), the function 𝑣(𝑥,𝑡) represented by (32) satisfies the following conditions: lim_𝑡→0𝑣(𝑥,𝑡)=0;𝑣(𝑥,𝑡)⟶0if𝑥⟶+∞. (68)
Finally, it is necessary to show that the function 𝑣(𝑥,𝑡) meets boundary condition (24). To this end, we observe the next difference (here we will essentially use statement (35)) 𝑣(𝑥,𝑡)−𝑔₁(𝑡≤_𝑥+∞)𝐺(𝑥,𝜏)𝑔₁(𝑡−𝜏)−𝑔₁(𝑡+)𝑑𝜏_𝑥0𝐺(𝑥,𝜏)𝑔₁(𝑡−𝜏)−𝑔₁(𝑡)𝑑𝜏≤2sup_[]0,𝑇𝑔_1𝑥^+∞𝐺(𝑥,𝜏)𝑑𝜏+𝑔_1[]^{((𝜃/2)𝛼)}_𝑡,0,𝑇×^𝑥₀𝐺(𝑥,𝜏)𝜏^𝛼𝜃/2𝑑𝜏≡𝐴₁+𝐴₂. (69) Applying inequality (38) with 𝑡∶=𝑥, 𝜃∶=𝜃 to the term 𝐴₂, we get 𝐴₂≤const.𝑔_{1𝐶((2+𝜃)/2)𝛼([0,𝑇])}𝑥^𝛼𝜃/2. (70) To estimate the term 𝐴₁ in (69), we use the Wright formula (see [38] or (2.2.5) in [21]): −𝛽𝑧𝑊𝑧;𝛽,1−𝛽=𝑊𝑧,𝛽,0, (71) and rewrite the function 𝐺(𝑥,𝑡) as 𝐺()=𝛼𝑥,𝑡2√𝑎₀𝑥𝑡^{−1−(𝛼/2)}𝑊−𝑥√𝑎₀𝑡^−𝛼/2𝛼;−2𝛼;1−2. (72) After that, due to Lemmas 2.2.4 and 2.2.7 in [21] and representation (72), we have 0<𝐺(𝑥,𝑡)<const.𝑥𝑡^{−1−𝛼/2}. (73) Thus, 𝐴₁≤const.sup_[]0,𝑇𝑔_1𝑥^+∞𝑥𝜏^{−1−𝛼/2}𝑑𝜏≤const.sup_[]0,𝑇𝑔₁𝑥^1−𝛼/2. (74) Then, we joint estimates (69)–(74) and obtain 𝑣(𝑥,𝑡)−𝑔₁(𝑡𝑔)≤const._{1𝐶((𝜃/2)𝛼)𝛼([0,𝑇])}𝑥^1−𝛼/2+𝑥^𝛼𝜃/2⟶0if𝑥⟶0, (75) which means that lim_𝑥→0𝑣(𝑥,𝑡)=𝑔₁(𝑡).
Therefore, as it was written above, the function 𝑣(𝑥,𝑡) given by (32) is a solution of (22)–(24) in the case of (27). The uniqueness of this solution is proved like the arguments of Theorem 3.2 from [39].

Now we remove restriction (27). To this end, it is enough to consider the Cauchy problem: 𝜕^𝛼_𝑡𝑣(𝑥,𝑡)−𝑎₀𝜕²𝑣(𝑥,𝑡)𝜕𝑥²=𝑔₀(𝑥,𝑡)in𝑅¹×(0,𝑇),𝛼∈(0,1);𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈𝑅¹;𝑣(𝑥,𝑡)⟶0,if|𝑥|⟶+∞,∀𝑡≥0. (76) Here, 𝑔₀(𝑥,𝑡) and 𝑣₀(𝑥) are extensions of the functions 𝑔₀(𝑥,𝑡) and 𝑣₀(𝑥), correspondingly, onto 𝑥<0. These functions together with their corresponding derivatives have finite supports and 𝑔_{0𝐶𝜃,(𝜃/2)𝛼(𝑅1×[0,𝑇])}≤𝐶𝑔_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇),𝑣_{0𝐶2+𝜃(𝑅1)}≤𝐶𝑣_{0𝐶2+𝜃(𝐷}⁺₎. (77) The results of Theorem 3.2 from [39] give the one-valued solvability of (76) and ‖𝑣‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(𝑅¹_×[0,𝑇])≤𝐶𝑔_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)+𝑣_{0𝐶2+𝜃(𝐷}⁺₎, (78) or, due to inequalities (77), ‖𝑣‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(𝑅¹_×[0,𝑇])≤𝐶𝑔_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)+𝑣_{0𝐶2+𝜃(𝐷}⁺₎. (79) After that, we will look for the solution 𝑣(𝑥,𝑡) of problem (22)–(24) as 𝑣(𝑥,𝑡)=𝑣(𝑥,𝑡)+𝑣(𝑥,𝑡),(𝑥,𝑡)∈𝐷⁺_𝑇, (80) where 𝑣(𝑥,𝑡) satisfies conditions (22)–(24) with the new right-hand sides which meet requirements of Lemma 8. Hence, we can apply the results of Lemma 8 to the function 𝑣(𝑥,𝑡). This fact and the properties (see (77)) of the function 𝑣(𝑥,𝑡) allow us to obtain the next results.

Theorem 9. Let 𝛼,𝜃∈(0,1), and conditions (25) and (26) hold, and 𝑔₀∈𝐶^{𝜃,(𝜃/2)𝛼}(𝐷⁺_𝑇), 𝑣₀∈𝐶^2+𝜃(𝐷⁺), 𝑔₁(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝑇]) for any positive number 𝑇. Then there exists a unique solution 𝑣∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇) of problem (22)–(24) and ‖𝑣‖_{𝐶2+𝜃,((2+𝜃)/2)𝛼(𝐷}⁺_𝑇)≤𝐶𝑔_{1𝐶((2+𝜃)/2)𝛼([0,𝑇])}+𝑔_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)+𝑣_{0𝐶2+𝜃(𝐷}⁺₎. (81)

3.2. The Model Initial-Boundary Value Problem with a Fractional Dynamic Boundary Condition

Here we study the following problem: it is necessary to find the function 𝑤(𝑥,𝑡) by the following conditions: 𝜕^𝛼_𝑡𝑤(𝑥,𝑡)−𝑎₀𝜕²𝑤(𝑥,𝑡)𝜕𝑥²=Φ₀(𝑥,𝑡)in𝐷⁺_𝑇(),𝑤(),𝛼∈0,1𝑥,0=𝑤₀(𝑥),𝑥∈𝐷⁺;𝑤()[],𝜕𝑥,𝑡⟶0if𝑥⟶+∞,∀𝑡∈0,𝑇_𝛼𝑡𝑤(0,𝑡)−𝑏₀𝜕𝑤(0,𝑡)𝜕𝑥=Φ₁(𝑡),𝑡∈[0,𝑇], (82) where Φ_𝑖,𝑖=0,1,𝑤₀ are the given functions. Let the following conditions hold: 𝑏₀𝜕𝑤₀(0)𝜕𝑥+Φ₁(0)=𝑎₀𝜕²𝑤₀(0)𝜕²𝑥+Φ₀();𝑤0,0₀(𝑥),Φ₀(𝑥,𝑡)≡0,if|𝑥|<𝑅₀∀𝑡∈[0,𝑇]; (83) for some positive number 𝑅₀.

At the beginning, we assume that 𝑤₀(𝑥),Φ₀(𝑥,𝑡)≡0 (84) and search a solution of (82) under this restriction in the class 𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇), if Φ₁(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]), 𝜃∈(0,1).

Note that conditions (83) and (84) allow us to extend the right-hand sides in (82) by 0 for 𝑡≤0. We save, for simplicity, the same notation for the extended function Φ₁(𝑡).

After the application of the Laplace transformation in time 𝑡 to problem (82), we have the following: 𝑝^𝛼𝑤𝑥,𝑝−𝑎₀𝜕²𝑤𝑥,𝑝𝜕𝑥²=0,𝑥∈𝐷⁺;𝑤𝑝𝑥,𝑝⟶0if𝑥⟶+∞;_𝛼𝑤0,𝑝−𝑏₀𝜕𝑤0,𝑝=Φ𝜕𝑥¹𝑝. (85) Here, we used again formula (30). Some simple calculations lead to the function 𝑤=Φ𝑥,𝑝¹𝑝𝑝^𝛼/2𝑝^𝛼/2+𝑏₀𝑎₀^−1/2exp−𝑥𝑎₀^−1/2𝑝^𝛼/2≡𝐹𝑝exp−𝑥𝑎₀^−1/2𝑝^𝛼/2, (86) which is the solution of problem (85). Due to formulas (2.30) in [34] and (1.80) in [29], we obtain, after applying of the inverse Laplace transformation to (86), that 𝑤(𝑥,𝑡)=^𝑡₀𝐺(𝑥,𝑡−𝜏)𝐹(𝜏)𝑑𝜏, (87) where the kernel 𝐺(𝑥,𝑡) is given by (33) and 𝐹(𝑡)=^𝑡₀𝐺₂(𝜏)Φ₁()𝐺𝑡−𝜏𝑑𝜏,₂(𝑡)=𝑡^𝛼−1𝐸_𝛼/2,𝛼−𝑏₀𝑎₀^−1/2𝑡^𝛼/2. (88) Here, 𝐸_𝛾,𝛽(𝑧) is the function of the Mittag-Leffler type, which is defined by the series expansion (see, e.g., (1.56) in [29] or (1.8.17) in [1]) 𝐸_𝛾,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ𝛾𝑘+𝛽,Re𝛾>0,𝛽,𝑧∈ℂ. (89) Note that this two-parameter function of the Mittag-Leffler type was in fact introduced by Agarwal [40].

The function 𝐺(𝑥,𝑡) has been studied in Section 3.1 (see Lemma 6). Thus, to describe the properties of the function 𝑤(𝑥,𝑡), we have to observe the function 𝐹(𝑡). To this end, we will use the following properties of the kernel 𝐺₂(𝑡), which are proved in Appendix B.

Lemma 10. Let 𝑁, 𝛿, and 𝜃 be some positive constants, 𝜃∈(0,1); 𝐼^𝜈_𝑡 and 𝐷^𝜈_𝑡 be the fractional Riemann-Liouville integral and derivative, correspondingly (their definitions are given in (49) and (47)). Then the following is true. (i)0≤𝐺₂(𝑡)𝑡≤𝐶^𝛼−11+𝑡^𝛼/2,^𝑡₀𝐺₂(𝜏)𝑑𝜏≤𝐶𝑡^𝛼.(ii)lim_𝑡→0𝐼_𝑡^1−𝛼𝐺₂(𝑡)()=1.iii₀^+∞𝐷^𝛼_𝜏𝐺₂(𝜏)()𝑑𝜏=−1.iv^𝑡₀𝐷^𝛼_𝜏𝐺₂(𝜏)𝜏^{(1+𝜃)𝛼/2}𝑑𝜏≤𝐶𝑡^{(1+𝜃)𝛼/2}.(v)_𝑁^+∞𝐷^𝛼_𝜏𝐺₂(𝜏()𝑑𝜏≤𝐶.vi)_𝛿^+∞𝐷^𝛼_𝜏𝐺₂(𝜏)𝐶𝑑𝜏≤1+1+𝛿^𝛼/2.(vii)_𝛿^+∞𝑑𝐷𝑑𝜏^𝛼𝜏𝐺₂(𝜏)𝜏^{(1+𝜃)𝛼/2}𝑑𝜏≤𝐶𝛿^{−1+𝛼(𝜃+1)/2}. (90)(91)(92)(93)(94)(95)(96)

Lemma 11. Let Φ₁(𝑡) meet requirement (83) and let Φ₁∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]). Then the function 𝐹(𝑡) represented by (88) belongs to 𝐶^{((3+𝜃)/2)𝛼}([0,𝑇]), 𝐹(0)=0, and the following estimate holds: ‖𝐹‖_𝐶^{((3+𝜃)/2)𝛼}_([0,𝑇])≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}. (97)

Proof. First of all, we evaluate the value of max_[0,𝑇]|𝐹(𝑡)|. To this end, we use the first inequality in (90). Thus, one has max_[0,𝑇]|𝐹(𝑡)|≤max_[]0,𝑇Φ₁(𝑡)^𝑡₀𝐺₂(𝜏)𝑑𝜏≤𝐶max_[]0,𝑇Φ₁(𝑡)𝑡^𝛼. (98) This inequality gives that 𝐹(0)=0. (99) Next, we obtain the representation of 𝜕^𝛼_𝑡𝐹(𝑡). Due to equality (99) and properties (46) and (48), we conclude that 𝜕^𝛼_𝑡𝐹(𝑡)=𝐷^𝛼_𝑡𝐹(𝑡)=Φ₁(𝑡)lim_𝑡→0𝐼_𝑡^1−𝛼𝐺₂(𝑡)+^𝑡₀Φ₁(𝑡−𝜏)𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏. (100) Since Φ₁(0)=0, we can rewrite the last equality as 𝜕^𝛼_𝑡𝐹(𝑡)=Φ₁(𝑡)lim_𝑡→0𝐼_𝑡^1−𝛼𝐺₂(𝑡)+₀^+∞Φ₁(𝑡−𝜏)𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏, (101) or, applying (91) and (92), we have 𝜕^𝛼_𝑡𝐹(𝑡)=₀^+∞Φ₁(𝑡−𝜏)−Φ₁(𝑡)𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏. (102) To estimate max_[0,𝑇]|𝜕^𝛼_𝑡𝐹(𝑡)|, we use representation (102) and get 𝜕^𝛼_𝑡𝐹(𝑡)≤^𝑁₀𝜏^{(1+𝜃)𝛼/2}𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏Φ_1[]^{((1+𝜃/2)𝛼)}_𝑡,0,𝑇+2max_[]0,𝑇Φ₁(𝑡)_𝑁^+∞𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏. (103) After that, we apply inequality (93) with 𝑡∶=𝛿 to the first term in (103) and (94) to the second and get max_[]0,𝑇𝜕^𝛼_𝑡𝐹(𝑡)≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}. (104) Finally, to complete the proof of Lemma 11 is necessary to estimate of ⟨𝜕^𝛼_𝑡𝐹⟩^{(((1+𝜃)/2)𝛼)}_{𝑡,[0,𝑇]}. Let 𝑡₂,𝑡₁∈[0,𝑇] and 𝑡₂>𝑡₁. Denote Δ𝑡∶=𝑡₂−𝑡₁,Δ_𝑡𝜕^𝛼_𝑡𝐹=𝜕^𝛼_𝑡𝐹𝑡₂−𝜕^𝛼_𝑡𝐹𝑡₁. (105)
Using formula (102), we represent the difference Δ_𝑡𝜕^𝛼_𝑡𝐹 as Δ_𝑡𝜕^𝛼_𝑡𝐹=^𝑡2_{𝑡1−Δ𝑡}𝐷^𝛼_𝜏𝐺₂𝑡₂−𝜏Φ₁(𝜏)−Φ₁𝑡₂+𝑑𝜏_{𝑡1𝑡1−Δ𝑡}𝐷^𝛼_𝜏𝐺₂𝑡₁−𝜏Φ₁𝑡₁−Φ₁(𝜏+)𝑑𝜏_{𝑡1−Δ𝑡−∞}𝐷^𝛼_𝜏𝐺₂𝑡₂−𝜏−𝐷^𝛼_𝜏𝐺₂𝑡₁×Φ−𝜏₁(𝜏)−Φ₁𝑡₂𝑑𝜏+∫^{𝑡1−Δ𝑡}_−∞𝐷^𝛼_𝜏𝐺₂𝑡₁−𝜏𝑑𝜏Φ₁𝑡₁−Φ₁𝑡₂≡⁴_𝑖=1𝔅_𝑖. (106) Changing the variable 𝑡₂−𝜏=𝑧 in 𝔅₁ and 𝑡₁−𝜏=𝑧 in 𝔅₂, we get 𝔅_𝑖=₀^2Δ𝑡𝐷^𝛼_𝜏𝐺₂(𝑧)Φ₁𝑡_𝑖−𝑧−Φ₁𝑡_𝑖𝑑𝑧,𝑖=1,2. (107) Then, the property of the function Φ₁(𝑡) together with inequality (93) (where 𝑡∶=2Δ𝑡) lead to 𝔅₁+𝔅₂≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}(Δ𝑡)^{(1+𝜃)𝛼/2}. (108) To get the same estimate for the term 𝔅₃, we apply the mean-value theorem to the difference [𝐷^𝛼_𝜏𝐺₂(𝑡₂−𝜏)−𝐷^𝛼_𝜏𝐺₂(𝑡₁−𝜏)] and inequality (96) and have after some simple calculations 𝔅₃≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}(Δ𝑡)^{(1+𝜃)𝛼/2}. (109) As for the estimate of 𝔅₄, one follows immediately from (95) where 𝛿∶=Δ𝑡 and properties of the function Φ₁ are 𝔅₄≤const.(Δ𝑡)^{(1+𝜃)𝛼/2+∞}_Δ𝑡𝐷^𝛼_𝜏𝐺₂(𝜏)Φ𝑑𝜏≤𝐶_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}(Δ𝑡)^{(1+𝜃)𝛼/2}. (110) Hence, inequalities (106)–(110) lead to estimate 𝜕^𝛼_𝑡𝐹_[]^{(((1+𝜃)/2)𝛼)}_𝑡,0,𝑇≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}(Δ𝑡)^{(1+𝜃)𝛼/2}, (111) which completes the proof of Lemma 11.

Due to results of this lemma and arguments like (70)–(74), we can infer that 𝑤(𝑥,𝑡)|_𝜕𝐷⁺_𝑇=𝐹(𝑡),𝜕^𝛼_𝑡𝑤(𝑥,𝑡)_{𝜕𝐷+𝑇}=𝜕^𝛼_𝑡𝐹(𝑡),𝑤(𝑥,𝑡)∈𝐶^{((3+𝜃)/2)𝛼}𝜕𝐷⁺_𝑇. (112) Based on the results of Lemma 11, properties of the function 𝐺(𝑥,𝑡) (see Lemmas 7 and 8), and (112), we can get the next assertion.

Lemma 12. Let conditions (83) and (84) hold and let Φ₁(𝑡) meet the requirements of Lemma 11. Then the function 𝑤(𝑥,𝑡) represented by (87) satisfies (112) and ‖𝑤‖_𝐶^{((3+𝜃)/2)𝛼}_(𝜕𝐷⁺_𝑇)+𝜕^𝛼_𝑡𝑤_{𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)≤𝐶Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}. (113)

Now, we can prove the solvability of problem (82).

Theorem 13. Let 𝜃,𝛼∈(0,1), and conditions (83) hold, and Φ₀(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(𝐷⁺_𝑇), 𝑤₀(𝑥)∈𝐶^2+𝜃(𝐷⁺), Φ₁(𝑡)∈𝐶^{((1+𝜃)/2)𝛼}([0,𝑇]) for any positive number 𝑇. Then there exists a unique solution 𝑤(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇) of problem (82) and ‖𝑤‖_{𝐶2+𝜃,((2+𝜃)/2)𝛼(𝐷}⁺_𝑇)+𝜕^𝛼_𝑡𝑤_{𝐶((1+𝜃)/2)𝛼(𝜕𝐷+𝑇)}≤𝐶Φ_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)+Φ_{1𝐶((1+𝜃)/2)𝛼([0,𝑇])}+𝑤_{0𝐶2+𝜃(𝐷}⁺₎. (114)

Proof. To prove this theorem in the case of (84) it is enough to consider the following Dirichlet problem: 𝜕^𝛼_𝑡𝑤(𝑥,𝑡)−𝑎₀𝜕²𝑤(𝑥,𝑡)𝜕𝑥²=0in𝐷⁺_𝑇();𝑤(),𝛼∈0,1𝑥,0=0,𝑥∈𝐷⁺;𝑤()[];𝑤()(𝑡)[],𝑥,𝑡⟶0if𝑥⟶+∞,∀𝑡∈0,𝑇0,𝑡=𝐹,𝑡∈0,𝑇 (115) and to apply the results of Theorem 9 and Lemma 12.
To remove restriction (84), we will look for the solution of problem (82) in the following form: 𝑤(𝑥,𝑡)=𝑤₁(𝑥,𝑡)+𝑤₂(𝑥,𝑡), (116) where 𝑤₁(𝑥,𝑡) is the solution of the following Dirichlet problem: 𝜕^𝛼_𝑡𝑤₁(𝑥,𝑡)−𝑎₀𝜕²𝑤₁(𝑥,𝑡)𝜕𝑥²=Φ₀(𝑥,𝑡)in𝐷⁺_𝑇();𝑤,𝛼∈0,1₁(𝑥,0)=𝑤₀(𝑥),𝑥∈𝐷⁺;𝑤₁()[];𝑤𝑥,𝑡⟶0if𝑥→+∞,∀𝑡∈0,𝑇₁(0,𝑡)=0,𝑡∈[0,𝑇] (117) and the function 𝑤₂(𝑥,𝑡) solves the following problem: 𝜕^𝛼_𝑡𝑤₂(𝑥,𝑡)−𝑎₀𝜕²𝑤₂(𝑥,𝑡)𝜕𝑥²=0in𝐷⁺_𝑇();𝑤,𝛼∈0,1₂(𝑥,𝑡)⟶_𝑥→+∞[]𝑤0∀𝑡∈0,𝑇₂(𝑥,0)=0,𝑥∈𝐷⁺;𝜕^𝛼_𝑡𝑤₂(0,𝑡)−𝑏₀𝜕𝑤₂(0,𝑡)𝜕𝑥=Φ₁(𝑡)+𝑏₀𝜕𝑤₁(0,𝑡)≡𝜕𝑥Φ₁(𝑡),𝑡∈[0,𝑇]. (118) The one-valued solvability of problem (117) follows from Theorem 9, which gives 𝑤_{1𝐶2+𝜃,((2+𝜃)/2)𝛼(𝐷}⁺_𝑇)+𝜕^𝛼_𝑡𝑤_{1𝐶((1+𝜃)/2)𝛼(𝜕𝐷+𝑇)}≤𝐶Φ_{0𝐶𝜃,(𝜃/2)𝛼(𝐷}⁺_𝑇)+𝑤_{0𝐶2+𝜃(𝐷}⁺₎,𝜕^𝛼_𝑡𝑤₁(0,𝑡)=0,𝑡∈[0,𝑇]. (119) These properties of the function 𝑤₁(𝑥,𝑡) allow us to conclude that Φ₁(𝑡)∈𝐶^((1+𝜃)/2)([0,𝑇]) and Φ₁(0)=0; that is, Φ₁(𝑡) satisfies the conditions of Lemma 12, and the right-hand sides of (118) meet requirement (84). It means that there exists a unique solution 𝑤₂∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(𝐷⁺_𝑇) of (118) which satisfies to inequality (119) with Φ₀(𝑥,𝑡)=𝑤₀(𝑥)≡0.
Finally, returning to representation (116) for 𝑤(𝑥,𝑡) and using the described properties of 𝑤_𝑖(𝑡,𝑥), 𝑖=1,2, we complete the proof of Theorem 13.

4. The Proofs of Theorems 3–5

Note that the proofs of Theorems 4 and 5 are analogous to the one of Theorem 3 and use the technique from Chapter 4 [32] together with the results of the solvability to the model problems from Section 3. That is why, we represent here only the proof of Theorem 3. In this route, we will need the solvability in 𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇) of the next initial-boundary value problem: 𝜕^𝛼_𝑡𝑈(𝑥,𝑡)−𝑎₀()𝜕𝑥,𝑡²𝑈(𝑥,𝑡)𝜕𝑥²+𝑎₁()()𝑥,𝑡𝜕𝑈𝑥,𝑡𝜕𝑥+𝑎₂(𝑥,𝑡)𝑈(𝑥,𝑡)=𝑔₀(),()𝑥,𝑡𝑥,𝑡∈Ω_𝑇(];𝑈(),𝛼∈0,1𝑥,𝑡=𝑔_𝑖(𝑡)onΓ_𝑖𝑇𝑈(),𝑖=1,2;𝑥,0=𝑈₀(𝑥)inΩ. (120) Reaching this goal is enough to repeat the arguments of Section 3.3 from [39] and apply the results of Theorem 9. Thus, we can assert the following.

Theorem 14. Let 𝜃,𝛼∈(0,1) and the following conditions 𝑔₁(0)=𝑈₀(𝑎);𝜕^𝛼_𝑡𝑔₁(0)=𝑎₀()𝜕𝑎,0²𝑈₀(𝑎)𝜕𝑥²−𝑎₁(𝑎,0)𝜕𝑈₀(𝑎)𝜕𝑥−𝑎₂(𝑎,0)𝑈₀(𝑎)+𝑔₀();𝑔𝑎,0₂(0)=𝑈₀(𝑏);𝜕^𝛼_𝑡𝑔₂(0)=𝑎₀()𝜕𝑏,0²𝑈₀(𝑏)𝜕𝑥²−𝑎₁(𝑏,0)𝜕𝑈₀(𝑏)𝜕𝑥−𝑎₂(𝑏,0)𝑈₀(𝑏)+𝑔₀(𝑏,0) (121) hold. Let and 𝑔₀(𝑥,𝑡)∈𝐶^{𝜃,(𝜃/2)𝛼}(Ω_𝑇), 𝑈₀(𝑥)∈𝐶^2+𝜃(Ω), 𝑔_𝑖(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝑇]), 𝑖=1,2, for any positive number 𝑇. Then there exists a unique solution 𝑢(𝑥,𝑡) of problem (120): 𝑈(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), and ‖𝑈‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)≤𝐶𝑔_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝑈_{0𝐶2+𝜃(Ω)}+²_𝑖=1𝑔_{𝑖𝐶((2+𝜃)/2)𝛼([0,𝑇])}, (122) where a positive constant 𝐶 depends only on the measure of Ω and ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇), 𝑖=0,2.

At the beginning, we prove Theorem 9 in the case of 𝑡∈[0,𝜏] for some small 𝜏<𝜏₀. Then, we will show how the solution 𝑢(𝑥,𝑡) is extended from [0,𝜏] into [0,𝑇], where 𝑇=𝜏+𝜏₀, for all 𝜏₀>0.

Lemma 15. Let the conditions of Theorem 3 hold. Then there exists a unique solution 𝑢(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝜏) of problem (1)–(5), for any 𝜏∈(0,𝜏₀], and ‖𝑢‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝜏)+𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝜏)}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝜏)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{1𝐶((1+𝜃)/2)𝛼(Γ2𝜏)}+𝑓_{2𝐶((2+𝜃)/2)𝛼(Γ1𝜏)}, (123) where the positive constant 𝐶 depends on ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝜏), 𝑖=0,2, ‖𝑏_𝑗‖_{𝐶((1+𝜃)/2)𝛼(Γ2𝜏)}, 𝑗=0,1, and the measure of Ω.

Proof. If the right-hand sides of (1)–(5) meet the following requirements: 𝑓₀(𝑥,𝑡)∈𝐶₀^{𝜃,(𝜃/2)𝛼}Ω_𝜏,𝑓₁(𝑡)∈𝐶₀^{((1+𝜃)/2)𝛼}Γ_2𝜏,𝑓₂(𝑡)∈𝐶₀^{((2+𝜃)/2)𝛼}Γ_1𝜏,𝑢₀(𝑥)≡0; (124) then by repeating the arguments of §4–§7 from Chapter 4 [32] and using the results of Theorems 9 and 13 and Theorem 3.2 from [39], we have proved the assertion of Lemma 15.
To remove restriction (124), we look for the solution of problem (1)–(5) as 𝑢(𝑥,𝑡)=𝑈(𝑥,𝑡)+𝑉(𝑥,𝑡), (125) where the function 𝑈(𝑥,𝑡) is the solution of problem (120) with 𝑔₀(𝑥,𝑡)∶=𝑓₀(𝑥,𝑡);𝑈₀(𝑥)∶=𝑢₀(𝑥);𝑔₁(𝑡)∶=𝑢₀(𝑎)+𝑈₁(𝑎)𝑡^𝛼(𝛼);𝑔𝛼Γ₂(𝑡)∶=𝑢₀(𝑏)+𝑈₁(𝑏)𝑡^𝛼(𝛼);𝑈𝛼Γ₁(𝑥)=𝑓₀(𝑥,0)+𝑎₀()𝜕𝑥,0²𝑢₀(𝑥)𝜕𝑥²−𝑎₁(𝑥,0)𝜕𝑢₀(𝑥)𝜕𝑥−𝑎₂(𝑥,0)𝑢₀(𝑥) (126) and the unknown function 𝑉(𝑥,𝑡) is the solution of the problem 𝜕^𝛼_𝑡𝑉(𝑥,𝑡)−𝑎₀()𝜕𝑥,𝑡²𝑉(𝑥,𝑡)𝜕𝑥²+𝑎₁()()𝑥,𝑡𝜕𝑉𝑥,𝑡𝜕𝑥+𝑎₂(𝑥,𝑡)𝑉(𝑥,𝑡)=0inΩ_𝜏;𝑉()𝑉()𝑥,0=0inΩ;𝑥,𝑡=𝑓₂(𝑡)−𝑔₁(𝑡)≡𝑓^∗₂(𝑡)onΓ_1𝜏;𝜕^𝛼_𝑡𝑉(𝑥,𝑡)+𝑏₀(𝑡)𝜕𝑉(𝑥,𝑡)𝜕𝑥+𝑏₁(𝑡)𝑉(𝑥,𝑡)=𝑓₁(𝑡)−𝜕^𝛼_𝑡𝑔₂(𝑡)−𝑏₀(𝑡)𝜕𝑈(𝑏,𝑡)𝜕𝑥−𝑏₁(𝑡)𝑔₂(𝑡)≡𝑓^∗₁(𝑡)onΓ_2𝜏. (127)
Our further arguments are divided into two parts. In the first step, we will show that the right-hand sides of (126) meet the requirements of Theorem 14, which will ensure the existence of the unique solution 𝑈(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝜏). The next step deals with the proving of the following equalities: 𝑓^∗_𝑖(0)=0,𝑖=1,2. (128)
We obtain after simple calculations the following properties for the functions 𝑔_𝑖(𝑡), for all 𝑡∈[0,𝜏]: 𝑔₁(0)=𝑢₀(𝑎),𝑔₂(0)=𝑢₀(𝑏);𝜕^𝛼_𝑡𝑔₁(𝑡)=𝑈₁(𝑎),𝜕^𝛼_𝑡𝑔₂(𝑡)=𝑈₂(𝑏);𝑔_𝑖(𝑡)∈𝐶^{((2+𝜃)/2)𝛼}([0,𝜏]),𝑖=1,2. (129)
After that, using (129), one can easily check that the right-hand sides in (126) satisfy conditions of Theorem 14. That is why, there exists a unique solution 𝑈(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝜏) as follows as follows: ‖𝑈‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝜏)≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝜏)}+𝑢_{0𝐶2+𝜃(Ω)}(),𝜕𝑈𝑏,0=𝜕𝑥𝜕𝑢₀(𝑏),𝜕𝑥 (130)(131) where the positive constant 𝐶 depends on ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝜏), 𝑖=0,2, and the measure of Ω.
Then we return to problem (127). Using properties (129) and (131), we get that 𝑓^∗₂(0)=𝑓₂(0)−𝑢₀(𝑎);𝑓^∗₁(0)=𝑓₁(0)−𝑏₀(0)𝜕𝑢₀(𝑏)𝜕𝑥−𝑏₁(0)𝑢₀(𝑏)−𝑓₀(𝑏,0)+𝑎₂(𝑏,0)𝑢₀(𝑏)−𝑎₀()𝜕𝑏,0²𝑢₀(𝑏)𝜕𝑥²+𝑎₁(𝑏,0)𝜕𝑢₀(𝑏).𝜕𝑥 (132) After that, to get statement (128) is enough to apply condition (15) to the right-hand sides of equalities (132).
Finally, taking into account (129)–(131), we obtain the following statements: 𝑓^∗_{1𝐶((1+𝜃)/2)𝛼([0,𝜏])}+𝑓^∗_{2𝐶((2+𝜃)/2)𝛼([0,𝜏])}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝜏)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{1𝐶((1+𝜃)/2)𝛼([0,𝜏])}+𝑓_{2𝐶((2+𝜃)/2)𝛼([0,𝜏])}, (133) where the positive constant 𝐶 depends on ‖𝑎_𝑖‖_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝜏), 𝑖=0,2, ‖𝑏_𝑗‖_{𝐶((1+𝜃)/2)𝛼([0,𝜏])}, 𝑗=0,1, and the measure of Ω.
Therefore, the right-hand sides of (127) meet requirements (124). It means that there exists a unique solution 𝑉(𝑥,𝑡) of problem (127) and that 𝑉(𝑥,𝑡) satisfies to inequality (123) with the corresponding right-hand sides. This fact together with the properties of the function 𝑈(𝑥,𝑡) (see (130)) and representation (125) completes the proof of Lemma 15.

Now we will extend the obtained solution 𝑢(𝑥,𝑡),𝑡∈[0,𝜏], in Lemma 15, into [𝜏,𝑇]. In other words, we construct the function 𝑈(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), 𝜕^𝛼_𝑡𝑈(𝑥,𝑡)∈𝐶^{((1+𝜃)/2)𝛼}(Γ_2𝑇) as follows: 𝜕^𝛼_𝑡𝑈(𝑥,𝑡)−𝑎₀()𝜕𝑥,𝑡²𝑈(𝑥,𝑡)𝜕𝑥²+𝑎₁()𝜕𝑥,𝑡𝑈(𝑥,𝑡)𝜕𝑥+𝑎₂(𝑥,𝑡)𝑈(𝑥,𝑡)=𝑓₀(𝑥,𝑡)inΩ_𝑇;𝑈(𝑥,𝑡)=𝑓₂(𝑡)onΓ_1𝑇;𝜕^𝛼_𝑡𝑈(𝑥,𝑡)+𝑏₀(𝑡)𝜕𝑈(𝑥,𝑡)𝜕𝑥+𝑏₁(𝑡)𝑈(𝑥,𝑡)=𝑓₁(𝑡)onΓ_2𝑇;𝑈(𝑥,𝑡)=𝑢(𝑥,𝑡),𝑥∈[];𝜕Ω,𝑡∈0,𝜏_𝛼𝑡𝑈(𝑏,𝑡)=𝜕^𝛼_𝑡𝑢(𝑏,𝑡),𝑡∈[0,𝜏], (134) where the function 𝑢(𝑥,𝑡) is constructed in Lemma 15; that is, the function 𝑢(𝑥,𝑡): 𝑢(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝜏), 𝜕^𝛼_𝑡𝑢(𝑥,𝑡)∈𝐶^{((1+𝜃)/2)𝛼}(Γ_2𝜏), is the unique solution of (1)–(5) for 𝑡∈[0,𝜏], 𝜏<𝜏₀.

First, we assume that 𝑇=2𝜏 and consider problem (120) with the following right-hand sides: 𝑔₀(𝑥,𝑡)∶=𝑓₀(𝑥,𝑡);𝑈₀(𝑥)∶=𝑢₀(𝑥);𝑔₁(𝑡)∶=𝑓₂(𝑡);𝑔₂(𝑡)∶=𝑢₀(𝑏)+1Γ(𝛼)^𝑡₀𝜏^𝛼−1𝑔(𝑡−𝜏)𝑑𝜏, (135) where 𝑔(𝑡)=𝜕^𝛼_𝑡𝑢()[];𝜕𝑏,𝑡,if𝑡∈0,𝜏_𝛼𝑡𝑢(𝑏,𝜏),if𝑡>𝜏. (136) After some calculations, we can infer that 𝜕^𝛼_𝑡𝑔₂(𝑡)=𝑔(𝑡);𝑔₂(0)=𝑢₀(𝑏);𝜕^𝛼_𝑡𝑔₂(0)=𝑓₀(𝑏,0)+𝑎₀()𝜕𝑏,0²𝑢₀(𝑏)𝜕𝑥²−𝑎₁(𝑏,0)𝜕𝑢₀(𝑏)𝜕𝑥−𝑎₂(𝑏,0)𝑢₀(𝑏);𝑔_{2𝐶((3+𝜃)/2)𝛼([0,2𝜏])}≤𝐶𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝜏)}+𝑢_{0𝐶2+𝜃(Ω)}. (137) Due to the fact the functions 𝑓₀,𝑓₂, and 𝑢₀ satisfy the consistency conditions, properties (137) allow us to conclude that the right-hand sides (135) meet requirements of Theorem 14. This means that there exists a unique solution 𝑈(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇), 𝑇=2𝜏, and that ‖𝑈‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{2𝐶((2+𝜃)/2)𝛼([0,𝑇])}+𝜕^𝛼_𝑡𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝜏)},𝜕^𝛼_𝑡𝑈=𝑔(𝑡)onΓ_2𝑇;𝜕^𝛼_𝑡𝑈|_Γ2𝑇∈𝐶^{((1+𝜃)/2)𝛼};𝑈()()([0,𝑇])𝑥,𝑡=𝑢𝑥,𝑡inΩ_𝜏. (138) After that, we search the solution of problem (134) in the following form: 𝑈(𝑥,𝑡)=𝑈(𝑥,𝑡)+Θ(𝑥,𝑡), (139) where the new unknown function Θ(𝑥,𝑡) satisfies the following conditions: 𝜕^𝛼_𝑡Θ(𝑥,𝑡)−𝑎₀()𝜕𝑥,𝑡²Θ(𝑥,𝑡)𝜕𝑥²+𝑎₁()()𝑥,𝑡𝜕Θ𝑥,𝑡𝜕𝑥+𝑎₂(𝑥,𝑡)Θ(𝑥,𝑡)=0inΩ_2𝜏;𝜕^𝛼_𝑡Θ(𝑏,𝑡)+𝑏₀(𝑡)𝜕Θ(𝑏,𝑡)𝜕𝑥+𝑏₁(𝑡)Θ(𝑏,𝑡)=𝑓₁(𝑡)−𝑔(𝑡)−𝑏₀(𝑡)𝜕𝑈(𝑏,𝑡)𝜕𝑥−𝑏₁(𝑡)𝑈(𝑏,𝑡)≡Φ₂(𝑡),Θ()[];Θ()()()𝑎,𝑡=0,𝑡∈0,2𝜏𝑥,𝑡=𝑢𝑥,𝑡−𝑈𝑥,𝑡≡Φ₃(𝑥,𝑡)inΩ_𝜏;𝜕^𝛼_𝑡Θ(𝑏,𝑡)=𝜕^𝛼_𝑡𝑢(𝑏,𝑡)−𝑔(𝑡)≡Φ₄(𝑡),𝑡∈[0,𝜏]. (140) As it follows from (136) and (138), if 𝑡∈[0,𝜏] then Φ₃(𝑥,𝑡)=0,𝑥∈ΦΩ,₂(𝑡)=Φ₄(𝑡)Φ=0,_{2𝐶((1+𝜃)/2)𝛼([0,2𝜏])}≤𝐶𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω2𝜏)}+𝑢_{0𝐶2+𝜃(Ω)}+𝑓_{2𝐶((2+𝜃)/2)𝛼([0,2𝜏])}+𝑓_{1𝐶((1+𝜃)/2)𝛼([0,2𝜏])}. (141)

In problem (140), we introduce the new variable: 𝜎=𝑡−𝜏. Let Θ(𝑥,𝜎)∶=Θ(𝑥,𝜎+𝜏);𝑎_𝑖(𝑥,𝜎)∶=𝑎_𝑖(𝑥,𝜎+𝜏),𝑖=0,2𝑏_𝑗(𝑥,𝜎)∶=𝑏_𝑗(𝑥,𝜎+𝜏),𝑗=0,1;Φ₂(𝜎)∶=Φ₂(𝜎+𝜏), (142) where 𝜎∈[−𝜏,𝜏] and 𝑎_𝑖, 𝑏_𝑗 satisfy the conditions of Theorem 3. Due to (141), we have Φ₂(𝜎)=0,if𝜎∈[−𝜏,0],Φ₂(𝜎)∈𝐶^{((1+𝜃)/2)𝛼}([−𝜏,𝜏]), (143)(144) and therefore Θ(𝑥,𝜎)=0,if𝜎∈[−𝜏,0]. (145)

To rewrite problem (140) in the new variable 𝜎, we use formula (3.110) from [39] as follows: 𝜕^𝛼_𝑡Θ(𝑥,𝑡)=𝜕^𝛼_𝜎Θ(𝑥,𝜎). (146) Thus, problem (140) in the new variable can be rewritten as 𝜕^𝛼_𝜎Θ(𝑥,𝜎)−𝑎₀()𝜕𝑥,𝜎²Θ(𝑥,𝜎)𝜕𝑥²+𝑎₁()𝜕𝑥,𝜎Θ(𝑥,𝜎)+𝜕𝑥𝑎₂(𝑥,𝜎)Θ(𝑥,𝜎)=0inΩ_𝜏;𝜕^𝛼_𝜎Θ(𝑏,𝜎)+𝑏₀(𝜎)𝜕Θ(𝑏,𝜎)+𝜕𝑥𝑏₁(𝜎)Θ()=𝑏,𝜎Φ₂(𝜎),𝜎∈[0,𝜏];Θ(𝑎,𝜎)=0,𝜎∈[0,𝜏];Θ(𝑥,0)=𝜕^𝛼_𝜎Θ(𝑏,0)=0inΩ. (147) Then we apply Lemma 15 to problem (147) and get the one-to-one solvability in 𝐶^{2+Θ,((2+Θ)/2)𝛼}(Ω_𝜏), and the estimate like (123) holds. Returning to the old variable 𝑡=𝜎+𝜏, we obtain the unique solution of problem (140), Θ(𝑥,𝑡)∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_2𝜏), and Θ(𝑥,𝑡) satisfies inequality (123), where 𝜏∶=2𝜏.

Thus, representation (139) together with the properties (138) for the functions 𝑈(𝑥,𝑡) and the corresponding properties of Θ(𝑥,𝑡) allow us to construct the solution of problem (1)–(5) for 𝑡∈[𝜏,2𝜏], if there is the solution for 𝑡∈[0,𝜏]. Repeating this procedure, we prove the existence of the solution for 𝑡∈[0,𝑇] and obtain estimate (19). As for the uniqueness of the solution, it follows from inequality (19). This ends the proof of Theorem 3.

5. The Local Solvability of the Nonlinear Problem with a Fractional Dynamic Boundary Condition

In this section, we indicate how our results may be applied to the nonlinear problem: 𝜕^𝛼_𝑡𝑣−𝑎₀𝑣,𝑣_𝑥𝑣_𝑥𝑥=𝑎₁𝑣,𝑣_𝑥,(𝑥,𝑡)∈Ω_𝑇();𝑣(),𝛼∈0,1𝑥,0=𝑢₀(𝑥)𝜕inΩ;_𝛼𝑡𝑣−𝑏^𝑖₀(𝑣)𝑣_𝑥=𝑏^𝑗₁(𝑣)onΓ_𝑗𝑇,𝑗=1,2. (148) We require the following conditions on the functions 𝑎_𝑖,  𝑏^𝑗_𝑘, 𝑘, 𝑖=0,1; 𝑗=1,2: (i) There exist some positive constants 𝛿_𝑙,𝑙=0,7 such that 0<𝛿₀<𝑎₀𝑣,𝑣_𝑥<𝛿₁;0<𝛿₂<𝑏¹₀(𝑣)<𝛿₃;−𝛿₄<𝑏²₀(𝑣)<−𝛿₅𝑏<0;_𝑗1<𝛿₆𝑎,𝑗=1,2;₁𝑣,𝑣_𝑥≤𝛿₇. (149)(ii) Consistency conditions hold: 𝑎₀𝑢₀,𝑢_0𝑥𝑢_0𝑥𝑥+𝑎₁𝑢₀,𝑢_0𝑥=𝑏^𝑗₀𝑢₀𝑢_0𝑥+𝑏^𝑗₁𝑢₀onΓ_𝑗𝑇,𝑗=1,2. (150)(iii) There exist 𝜕𝑎_𝑖(𝑣,𝑣_𝑥)/𝜕𝑣, 𝜕𝑎_𝑖(𝑣,𝑣_𝑥)/𝜕𝑣_𝑥, 𝜕𝑏^𝑗_𝑘(𝑣)/𝜕𝑣, 𝑘,𝑖=0,1, 𝑗=1,2, such that for some positive constant 𝐿¹_𝑖=0𝜕𝑎_𝑖𝑣,𝑣_𝑥−𝜕𝑣𝜕𝑎_𝑖𝑢,𝑢_𝑥+𝜕𝑢𝜕𝑎_𝑖𝑣,𝑣_𝑥𝜕𝑣_𝑥−𝜕𝑎_𝑖𝑢,𝑢_𝑥𝜕𝑢_𝑥≤𝐿|𝑣−𝑢|+𝑣_𝑥−𝑢_𝑥;_2𝑗=1𝜕𝑏^𝑗₀(𝑣)−𝜕𝑣𝜕𝑏^𝑗₀(𝑢)+𝜕𝑢𝜕𝑏^𝑗₁(𝑣)−𝜕𝑣𝜕𝑏^𝑗₁(𝑢)||,𝜕𝑢≤𝐿𝑣−𝑢 (151) for any bounded functions 𝑢,𝑣 together with their derivatives. (iv) There are positive constants 𝐴_𝑖,𝐵_𝑗,  𝑖,𝑗=0,3 as follows: 𝜕𝑎₀𝑣,𝑣_𝑥𝜕𝑣≤𝐴₀;𝜕𝑎₁𝑣,𝑣_𝑥𝜕𝑣≤𝐴₁;𝜕𝑎₀𝑣,𝑣_𝑥𝜕𝑣_𝑥≤𝐴₂;𝜕𝑎₁𝑣,𝑣_𝑥𝜕𝑣_𝑥≤𝐴₃;𝜕𝑏¹₀(𝑣)𝜕𝑣≤𝐵₀;𝜕𝑏²₀(𝑣)𝜕𝑣≤𝐵₁;𝜕𝑏¹₁(𝑣)𝜕𝑣≤𝐵₂;𝜕𝑏²₁(𝑣)𝜕𝑣≤𝐵₃; (152)(v)𝑢₀(𝑥)∈𝐶^2+𝜃Ω. (153)We introduce the functional spaces ℍ and ℍ₀ as follows: ℍ=𝐶^{2+𝜃,((2+𝜃)/2)𝛼}Ω_𝑇×𝐶^{((1+𝜃)/2)𝛼}Γ_1𝑇×𝐶^{((1+𝜃)/2)𝛼}Γ_2𝑇ℍ₀=𝐶^{𝜃,(𝜃/2)𝛼}Ω_𝑇×𝐶^{((1+𝜃)/2)𝛼}Γ_1𝑇×𝐶^{((1+𝜃)/2)𝛼}Γ_2𝑇, (154) and for any elements 𝑧=(𝑣,𝜕^𝛼_𝑡𝑣∣_𝜕Ω𝑇)∈ℍ, and 𝔉(𝑧)=(Φ₀,Φ₁,Φ₂)∈ℍ₀, we denote ‖𝑧‖_ℍ∶=‖𝑣‖_𝐶^{2+𝜃,((2+𝜃)/2)𝛼}_(Ω𝑇)+𝜕^𝛼_𝑡𝑣_{𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+𝜕^𝛼_𝑡𝑣_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)};‖𝔉(𝑧)‖_ℍ0∶=Φ_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+Φ_{1𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+Φ_{2𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}. (155)

Let 𝐵_𝑟(𝑣)={𝑣∈ℍ∶‖𝑣‖_ℍ≤𝑟} and 𝐵⁰_𝑟(𝑣)={𝑣∈ℍ∶‖𝑣‖_ℍ≤𝑟,𝑣(𝑥,0)=0}, 𝑟<𝑟₀, be balls of radius 𝑟 in the space ℍ, centered at the origin, for some positive 𝑟₀ to be determined later on.

The simple calculations lead to the following assertion.

Proposition 16. Let conditions (149)–(153) hold and the function 𝑢∈𝐵_𝑟(𝑢); then 𝑎_𝑖(𝑢,𝑢_𝑥), 𝜕𝑎_𝑖(𝑢,𝑢_𝑥)/𝜕𝑢, 𝜕𝑎_𝑖(𝑢,𝑢_𝑥)/𝜕𝑢_𝑥∈𝐶^{𝜃,(𝜃/2)}(Ω_𝑇); 𝑏^𝑗_𝑘(𝑢), 𝜕𝑏^𝑗_𝑘/𝜕𝑢∈𝐶^((1+𝜃)/2)(Γ_𝑗𝑇); 𝑖,𝑘=0,1, 𝑗=1,2, and 𝑎_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}≤max𝛿₀,𝛿₁+𝐴₀+𝐴₂‖𝑢‖_ℍ1+𝑇^𝛼/2+𝑇^{((2−𝜃)/2)𝛼};𝑎_{1𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}≤𝛿₇+𝐴₁+𝐴₃‖𝑢‖_ℍ1+𝑇^𝛼/2+𝑇^{((2−𝜃)/2)𝛼};_2𝑗=1𝑏^𝑗_{0𝐶((1+𝜃)/2)𝛼(Γ𝑗𝑇)}≤max𝛿₂,𝛿₃+max𝛿₄,𝛿₅+𝐵₀+𝐵₁‖𝑢‖_ℍ𝑇^𝛼+𝑇^{((1+𝜃)/2)𝛼};𝑏¹_{1𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+𝑏²_{1𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤𝛿₆+𝐵₃+𝐵₂‖𝑢‖_ℍ𝑇^𝛼+𝑇^{((1+𝜃)/2)𝛼};𝜕𝑎₀𝜕𝑢_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇)+𝜕𝑎₀𝜕𝑢_𝑥𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇)≤2𝐿‖𝑢‖_ℍ1+𝑇^{((2−𝜃)/2)𝛼}+𝑇^𝛼/2+𝐴₀+𝐴₂;𝜕𝑎₁𝜕𝑢_𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇)+𝜕𝑎₁𝜕𝑢_𝑥𝐶^{𝜃,(𝜃/2)𝛼}_(Ω𝑇)≤2𝐿‖𝑢‖_ℍ1+𝑇^{((2−𝜃)/2)𝛼}+𝑇^𝛼/2+𝐴₁+𝐴₃;𝜕𝑏¹₀𝜕𝑢_{𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+𝜕𝑏²₀𝜕𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤2𝐿‖𝑢‖_ℍ1+𝑇^{((1−𝜃)/2)𝛼}+𝐵₀+𝐵₁;𝜕𝑏¹₁𝜕𝑢_{𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+𝜕𝑏²₁𝜕𝑢_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤2𝐿‖𝑢‖_ℍ1+𝑇^{((1−𝜃)/2)𝛼}+𝐵₃+𝐵₂. (156) Moreover, for the functions 𝑝_𝜎(𝑢,𝑣)∶=𝜎𝑢+(1−𝜎)𝑣 and 𝑞_𝜎(𝑢,𝑣)∶=𝜎𝑢_𝑥+(1−𝜎)𝑣_𝑥, for all 𝜎∈[0,1], 𝑢,𝑣∈𝐵_𝑟(𝑢), the following inequalities hold: 𝜕𝑎_𝑖𝑝_𝜎(𝑢,v),𝑣_𝑥𝜕𝑝_𝜎(𝑢,𝑣)≤𝐿max_Ω𝑇|𝑢|+max_{𝑥,𝑡,𝑣,𝑣𝑥}𝜕𝑎_𝑖𝑣,𝑣_𝑥;𝜕𝑣𝜕𝑎_𝑖𝑣,𝑞_𝜎(𝑢,𝑣)𝜕𝑞_𝜎(𝑢,𝑣)≤𝐿max_Ω𝑇𝑢_𝑥+max_{𝑥,𝑡,𝑣,𝑣𝑥}𝜕𝑎_𝑖𝑣,𝑣_𝑥𝜕𝑣_𝑥;𝜕𝑏^𝑗_𝑖𝑝_𝜎(𝑢,𝑣)𝜕𝑝_𝜎(𝑢,𝑣)≤𝐿max_Ω𝑇|𝑢|+max_{𝑥,𝑡,𝑣}𝜕𝑏^𝑗_𝑖(𝑣),𝜕𝑣𝑖=0,1;𝑗=1,2. (157)

First, we linearize problem (148) on the initial data and represent one as a system 𝔄𝑧=𝔗(𝑧), where 𝔄 is a linear operator and 𝔗(𝑧) is a nonlinear perturbation. To this end, we introduce the new unknown function as follows: 𝑤(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑉₀(𝑥,𝑡), (158) where the function 𝑉₀(𝑥,𝑡) is a solution of the following problem: 𝜕^𝛼_𝑡𝑉₀−𝑎₀𝑢₀,𝑢_0𝑥𝑉_0𝑥𝑥=𝑎₁𝑢₀,𝑢_0𝑥,(𝑥,𝑡)∈Ω_𝑇();𝑉(),𝛼∈0,1𝑥,0=𝑢₀(𝑥)𝜕inΩ;_𝛼𝑡𝑉₀−𝑏^𝑖₀𝑢₀𝑉_0𝑥=𝑏^𝑗₁𝑢₀onΓ_𝑗𝑇,𝑗=1,2. (159) By Theorem 5 and Proposition 16, there exists a unique solution 𝑉₀(𝑥,𝑡) of the problem (159) and 𝑉_0ℍ≤𝐶₀𝑢_{0𝐶2+𝜃(Ω)}, (160) where the positive constant 𝐶₀ depends on 𝛿_𝑖, 𝑖=0,7, 𝐵_𝑘,𝐴_𝑗, 𝑘,𝑗=0,3, 𝐿.

Next, we rewrite problem (148) in terms of the function 𝑤(𝑥,𝑡) and after some calculations get the problem in the form: 𝜕^𝛼_𝑡𝑤−𝑎₀𝑉₀,𝑉_0𝑥𝑤_𝑥𝑥=Φ₀𝑤,𝑤_𝑥,𝑤_𝑥𝑥,(𝑥,𝑡)∈Ω_𝑇();𝑤()𝜕,𝛼∈0,1𝑥,0=0inΩ;_𝛼𝑡𝑤−𝑏^𝑖₀𝑉₀𝑤_𝑥=Φ^𝑗₁𝑤,𝑤_𝑥onΓ_𝑗𝑇,𝑗=1,2, (161) where Φ₀𝑤,𝑤_𝑥,𝑤_𝑥𝑥=𝑓₀(𝑥,𝑡)+𝑎₀𝑉₀+𝑤,𝑉_0𝑥+𝑤_𝑥−𝑎₀𝑉₀,𝑉_0𝑥×𝑤_𝑥𝑥+𝑉_0𝑥𝑥+𝑎₁𝑉₀+𝑤,𝑉_0𝑥+𝑤_𝑥−𝑎₁𝑉₀,𝑉_0𝑥,𝑓₀(𝑥,𝑡)=𝑎₀𝑉₀,𝑉_0𝑥−𝑎₀𝑢₀,𝑢_0𝑥𝑉_0𝑥𝑥+𝑎₁𝑉₀,𝑉_0𝑥−𝑎₁𝑢₀,𝑢_0𝑥,Φ^𝑗₁𝑤,𝑤_𝑥=𝑓^𝑗₁()+𝑏𝑥,𝑡_𝑗0𝑉₀+𝑤−𝑏^𝑗₀𝑉₀𝑤_𝑥+𝑉_0𝑥+𝑏^𝑗₁𝑉₀+𝑤−𝑏^𝑗₁𝑉₀,𝑓^𝑗₁(𝑥,𝑡)=𝑏^𝑗₀𝑉₀−𝑏^𝑗₀𝑢₀𝑉_0𝑥+𝑏^𝑗₁𝑉₀−𝑏^𝑗₁𝑢₀. (162)(163)(164)(165)

Thus, we represent nonlinear problem (148) as 𝔄𝑤=𝔗(𝑤). (166) Note that if we froze the functional arguments in the functions Φ₀(𝑤,𝑤_𝑥,𝑤_𝑥𝑥) and Φ^𝑗₁(𝑤,𝑤_𝑥), 𝑗=1,2, then problem (161) will be a linear problem with variable coefficients, which has been studied in detail in Sections 2–4. By Theorem 5, 𝔄 has a bounded inverse 𝔄⁻¹, so that 𝑤=𝔄⁻¹𝔗(𝑤)≡𝔖(𝑤), (167) and 𝔖 is a nonlinear operator. We will show that 𝔖 is a contraction operator.

Lemma 17. The following inequalities hold for the right-hand sides of problem (167): ‖𝔗(0)‖_ℍ0≤𝐶₁(𝑇),‖𝔗(𝑤)−𝔗(𝑣)‖_ℍ0≤𝐶₂(𝑇,𝑟)‖𝑤−𝑣‖_ℍ,∀𝑤,𝑣,∈𝐵⁰_𝑟(𝑢), (168)(169) with 𝐶₁(𝑇)→0𝐶₂(𝑇,𝑟)→0 as 𝑟,𝑇→0.

Proof. First we prove (168). Note that ‖𝔗(0)‖_ℍ0=𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+²_𝑗=1𝑓^𝑗_{1𝐶((1+𝜃)/2)𝛼(Γ𝑗𝑇)}. (170) Let 𝑝_𝜎∶=𝑝_𝜎𝑧₁,𝑧₂=𝜎𝑧₁(𝑥,𝑡)+(1−𝜎)𝑧₂();𝑞𝑥,𝑡_𝜎∶=𝑞_𝜎𝑧₁,𝑧₂=𝜎𝑧_1𝑥(𝑥,𝑡)+(1−𝜎)𝑧_2𝑥(𝑥,𝑡);∀𝑧_𝑖(𝑥,𝑡)∈𝐵_𝑟(𝑢),𝑖=1,2,𝜎∈[0,1]. (171) Using definitions (163), (165) and notations (171) with 𝑧₁∶=𝑉₀(𝑥,𝑡) and 𝑧₂∶=𝑢₀(𝑥), we have 𝑓₀(𝑥,𝑡)=𝑉_0𝑥𝑥¹₀𝜕𝑎₀𝑝_𝜎,𝑉_0𝑥𝜕𝑝_𝜎𝑑𝜎𝑉₀(𝑥,𝑡)−𝑢₀(𝑥)+𝑉_0𝑥𝑥¹₀𝜕𝑎₀𝑢₀,𝑞_𝜎𝜕𝑞_𝜎𝑑𝜎𝑉_0𝑥(𝑥,𝑡)−𝑢_0𝑥(𝑥+₁₀)𝜕𝑎₁𝑝_𝜎,𝑉_0𝑥𝜕𝑝_𝜎𝑑𝜎𝑉₀(𝑥,𝑡)−𝑢₀(𝑥+₁₀)𝜕𝑎₁𝑢₀,𝑞_𝜎𝜕𝑞_𝜎𝑑𝜎𝑉_0𝑥(𝑥,𝑡)−𝑢_0𝑥(𝑥;𝑓_𝑗1)(𝑥,𝑡)=𝑉_0𝑥¹₀𝜕𝑏^𝑗₀𝑝_𝜎𝜕𝑝_𝜎𝑑𝜎𝑉₀(𝑥,𝑡)−𝑢₀(𝑥)+¹₀𝜕𝑏^𝑗₁𝑝_𝜎𝜕𝑝_𝜎𝑑𝜎𝑉₀(𝑥,𝑡)−𝑢₀(𝑥). (172) It is easy to examine that the following are true: 𝑉₀(𝑥,𝑡)−𝑢₀(𝑥)≤𝑇^𝛼𝑉_0ℍ;𝑉_0𝑥(𝑥,𝑡)−𝑢_0𝑥(𝑥)≤𝑇^{((1+𝜃)/2)𝛼}𝑉_0ℍ;𝑉₀−𝑢₀^(𝜃)_𝑥,Ω𝑇≤𝑇^{((1+𝜃)/2)𝛼}𝑉_0ℍ;𝑉_0𝑥−𝑢_0𝑥^(𝜃)_𝑥,Ω𝑇≤𝑇^(𝜃/2)𝛼𝑉_0ℍ;𝑉₀−𝑢₀^{((𝜃/2)𝛼)}_𝑡,Ω𝑇≤𝑇^{((2−𝜃)/2)𝛼}𝑉_0ℍ;𝑉_0𝑥−𝑢_0𝑥^{((𝜃/2)𝛼)}_𝑡,Ω𝑇≤𝑇^𝛼/2𝑉_0ℍ. (173) Hence, using inequalities (173), representation (172), and results of Proposition 16, we get after some tedious calculations 𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}≤𝐶₁𝑇^𝜃𝛼/21+³_𝑖=0𝐴_𝑖×1+𝐿𝑢_{0𝐶2+𝜃(Ω)}1+𝑇^𝛼/2+𝑇^{(2−𝜃)𝛼/22};_2𝑗=1𝑓^𝑗_{1𝐶((1+𝜃)/2)𝛼(Γ𝑗𝑇)}≤𝐶₁max𝑇^𝜃/𝛼2,𝑇^{(1−𝜃)𝛼/2}1+³_𝑖=0𝐵_𝑖×1+𝐿𝑢_{0𝐶2+𝜃(Ω)}1+𝑇^{(1−𝜃)𝛼/22},𝐶₁=max1,𝐶₀, (174) where the constant 𝐶₀ is from (160).
Let 𝐶₁(𝑇)∶=𝐶₁max𝑇^𝜃/𝛼2,𝑇^{(1−𝜃)𝛼/2}×1+³_𝑖=0𝐵_𝑖+𝐴_𝑖×1+𝐿𝑢_{0𝐶2+𝜃(Ω)}1+𝑇^𝜃/𝛼2+𝑇^{(2−𝜃)𝛼/22}. (175) It is easy to see that 𝐶₁(𝑇)→0, as 𝑇→0. Thus inequalities (174) together with representation (170) lead to estimate (168), where 𝐶₁(𝑇) is given by (175).
Next, we will obtain inequality (169). Note that, as it follows from definitions (162) and (164), one has ‖𝔗(𝑤)−𝔗(𝑣)‖_ℍ0=Φ₀𝑤,𝑤_𝑥,𝑤_𝑥𝑥−𝑓₀−Φ₀𝑣,𝑣_𝑥,𝑣_𝑥𝑥−𝑓_{0𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+²_𝑗=1Φ^𝑗₁𝑤,𝑤_𝑥−𝑓^𝑗₁−Φ^𝑗₁𝑣,𝑣_𝑥−𝑓^𝑗_{1𝐶((1+𝜃)/2)𝛼(Γ𝑗𝑇)}. (176)
Denote 𝑙_𝜎=𝜎𝑤_𝑥𝑥(𝑥,𝑡)+(1−𝜎)𝑣_𝑥𝑥()[];𝑄𝑥,𝑡,𝜎∈0,1𝑧,𝑧_𝑥,𝑧_𝑥𝑥∶=𝑎₀V₀+𝑧,𝑉_0𝑥+𝑧_𝑥−𝑎₀𝑉₀,𝑉_0𝑥𝑧_𝑥𝑥+𝑉_0𝑥𝑥;𝑄^𝑗₁𝑧,𝑧_𝑥∶=𝑏^𝑗₀𝑉₀+𝑧−𝑏^𝑗₀𝑉₀𝑧_𝑥+𝑉_0𝑥,∀𝑧∈𝐵⁰_𝑟(𝑢). (177)
Then, we have the following: Φ₀𝑤,𝑤_𝑥,𝑤_𝑥𝑥−𝑓₀−Φ₀𝑣,𝑣_𝑥,𝑣_𝑥𝑥−𝑓₀=¹₀𝜕𝑄𝑝_𝜎,𝑤_𝑥,𝑤_𝑥𝑥𝜕𝑝_𝜎()+𝑑𝜎𝑤−𝑣₁₀𝜕𝑄𝑣,𝑞_𝜎,𝑤_𝑥𝑥𝜕𝑞_𝜎𝑑𝜎𝑤_𝑥−𝑣_𝑥+¹₀𝜕𝑄𝑣,𝑣_𝑥,𝑙_𝜎𝜕𝑙_𝜎𝑑𝜎𝑤_𝑥𝑥−𝑣_𝑥𝑥+()×𝑤−𝑣₁₀𝜕𝑎₁𝑝_𝜎,𝑉_0𝑥+𝑤_𝑥𝜕𝑝_𝜎+𝑑𝜎₁₀𝜕𝑎₁𝑉₀+𝑣,𝑞_𝜎𝜕𝑞_𝜎𝑑𝜎𝑤_𝑥−𝑣_𝑥;Φ^𝑗₁𝑤,𝑤_𝑥−𝑓^𝑗₁−Φ^𝑗₁𝑣,𝑣_𝑥−𝑓^𝑗₁=¹₀𝜕𝑄^𝑗₁𝑝_𝜎,𝑤_𝑥𝜕𝑝_𝜎()+𝑑𝜎𝑤−𝑣₁₀𝜕𝑄^𝑗₁𝑣,𝑞_𝜎𝜕𝑞_𝜎𝑑𝜎𝑤_𝑥−𝑣_𝑥+¹₀𝜕𝑏^𝑗₁𝑝_𝜎𝜕𝑝_𝜎𝑑𝜎(𝑤−𝑣). (178)
After that, the simple calculations together with the results of Proposition 16 allow us to get the following for any function 𝑧(𝑥,𝑡)∈𝐵⁰_𝑟(𝑢). (i)𝜕𝑄𝑧,𝑧_𝑥,𝑧_𝑥𝑥𝜕𝑧_{𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝜕𝑄𝑧,𝑧_𝑥,𝑧_𝑥𝑥𝜕𝑧_{𝑥𝐶𝜃,(𝜃/2)𝛼(Ω𝑇)}+𝜕𝑄¹₁𝑧,𝑧_𝑥𝜕𝑧_{𝐶((1+𝜃)/2)𝛼(Γ1𝑇)}+𝜕𝑄²₁𝑧,𝑧_𝑥𝜕𝑧_{𝐶((1+𝜃)/2)𝛼(Γ2𝑇)}≤‖𝑧‖_ℍ+𝐶₀𝑢_{0𝐶2+𝜃(Ω)}×2𝐿‖𝑧‖_ℍ+¹_𝑗=0𝐵_𝑗+𝐴₀+𝐴₂.(ii)max_Ω𝑇𝜕𝑄𝑧,𝑧_𝑥,𝑧_𝑥𝑥𝜕𝑧_𝑥𝑥≤𝑇^{(1+𝜃)𝛼/2}‖𝑧‖_ℍ2𝐿‖𝑧‖_ℍ+𝐴₀+𝐴₂1+𝑇^𝛼/2;𝜕𝑄𝑧,𝑧_𝑥,𝑧_𝑥𝑥𝜕𝑧_𝑥𝑥^{((𝜃/2)𝛼)}_𝑡,Ω𝑇≤𝑇^𝛼/2‖𝑧‖_ℍ2𝐿‖𝑧‖_ℍ+𝐴₀+𝐴₂×1+𝑇^𝜃𝛼/2+𝑇^{(1−𝜃)𝛼/2};𝜕𝑄𝑧,𝑧_𝑥,𝑧_𝑥𝑥𝜕𝑧_𝑥𝑥^(𝜃)_𝑥,Ω𝑇≤𝑇^{(1+𝜃)𝛼/2}‖𝑧‖_ℍ2𝐿‖𝑧‖_ℍ+𝐴₀+𝐴₂×1+𝑇^{(1−𝜃)𝛼/2}.(iii)max_Ω𝑇𝜕𝑄^𝑗₁𝑧,𝑧_𝑥𝜕𝑧_𝑥≤𝑇^𝛼‖𝑧‖_ℍ𝐿‖𝑧‖_ℍ+¹_𝑗=0𝐵_𝑗;𝜕𝑄^𝑗₁𝑧,𝑧_𝑥𝜕𝑧_𝑥^{(((1+𝜃)/2)𝛼)}_𝑡,Ω𝑇≤𝑇^{(1−𝜃)𝛼/2}1+𝑇^{(1+𝜃)𝛼/2}𝐿‖𝑧‖_ℍ+¹_𝑗=0𝐵_𝑗;𝜕𝑄^𝑗₁𝑧,𝑧_𝑥𝜕𝑧_𝑥^(𝜃)_𝑥,Ω𝑇≤𝑇^{(1+𝜃)𝛼/2}‖𝑧‖_ℍ𝐿‖𝑧‖_ℍ+¹_𝑗=0𝐵_𝑗1+𝑇^{(1−𝜃)𝛼/2}. (179)(180)(181)
Moreover, due to 𝑣,𝑤∈𝐵⁰_𝑟(𝑢), we have the following: 𝑇^−𝛼max_Ω𝑇|𝑣−𝑤|+𝑇^{−(1+𝜃)𝛼/2}max_Ω𝑇𝑣_𝑥−𝑤_𝑥+𝑇^{−𝜃𝛼/2}max_Ω𝑇𝑣_𝑥𝑥−𝑤_𝑥𝑥≤‖𝑤−𝑣‖_ℍ;𝑇^{−(1+𝜃)𝛼/2}⟨𝑣−𝑤⟩^(𝜃)_𝑥,Ω𝑇+𝑇^{−𝜃𝛼/2}𝑣_𝑥−𝑤_𝑥^(𝜃)_𝑥,Ω𝑇≤‖𝑤−𝑣‖_ℍ;𝑇^{−((2−𝜃)/2)𝛼}⟨𝑣−𝑤⟩^{((𝜃/2)𝛼)}_𝑡,Ω𝑇+𝑇^−𝛼/2𝑣_𝑥−𝑤_𝑥^{((𝜃/2)𝛼)}_𝑡,Ω𝑇≤‖𝑤−𝑣‖_ℍ. (182) Thus, applying inequalities from Proposition 16 and (179)–(182) to the right-hand sides in (178), we get (169), where 𝐶₂(𝑇,𝑟)=𝐶₁max𝑇^𝜃𝛼/2,𝑇^{(1−𝜃)𝛼/2}×𝐿𝑟+³_𝑖=0𝐵_𝑖+𝐴_𝑖1+𝑟+𝑢_{0𝐶2+𝜃(Ω)}×1+𝑇^3𝛼/2+𝑇^{(3+𝜃)𝛼/2}+𝑇^{𝛼+𝛼𝜃}. (183) It is obviously 𝐶₂(𝑇,𝑟)→0 as 𝑟,𝑇→0.

As it follows from Lemma 17, for sufficiently small 𝑇 and 𝑟₀, the nonlinear operator 𝔖 satisfies the conditions of the fixed point theorem for a contraction operator. Hence, we have proved the following theorem.

Theorem 18. Let conditions (149)–(153) hold. Then there exists a unique solution of problem (148) for small interval 0<𝑡<𝑇, such that 𝑣∈𝐶^{2+𝜃,((2+𝜃)/2)𝛼}(Ω_𝑇) and that 𝜕^𝛼_𝑡𝑣∈𝐶^{((1+𝜃)/2)𝛼}(Γ_𝑗𝑇), 𝑗=1,2.

Using the analogous arguments, it is possible to assert the following results.

Remark 19. If we exchange the dynamic boundary condition on Γ_1𝑇 by either (5) or (6) and assume that the corresponding consistency conditions hold, then the results of Theorem 18 save.

Remark 20. The results of Theorem 18 hold if conditions (152) are changed by 𝜕𝑎₀𝑣,𝑣_𝑥𝜕𝑣≤𝐴₀1+|𝑣|+𝑣_𝑥;𝜕𝑎₁𝑣,𝑣_𝑥𝜕𝑣≤𝐴₁1+|𝑣|+𝑣_𝑥;𝜕𝑎₀𝑣,𝑣_𝑥𝜕𝑣_𝑥≤𝐴₂1+|𝑣|+𝑣_𝑥;𝜕𝑎₁𝑣,𝑣_𝑥𝜕𝑣_𝑥≤𝐴₃1+|𝑣|+𝑣_𝑥;𝜕𝑏¹₀(𝑣)𝜕𝑣≤𝐵₀[1+|𝑣|];𝜕𝑏²₀(𝑣)𝜕𝑣≤𝐵₁[1+|𝑣|];𝜕𝑏¹₁(𝑣)𝜕𝑣≤𝐵₂[1+|𝑣|];𝜕𝑏²₁(𝑣)𝜕𝑣≤𝐵₃[1+|𝑣|] (184) and constants 𝛿₆ and 𝛿₇ in requirements (149) are replaced by 𝛿₆[1+|𝑣|] and 𝛿₇[1+|𝑣|+|𝑣_𝑥|], correspondingly.

Appendix

Here we give the proof of some estimates used in the arguments above.

A. The proof of Lemma 6

The statement (i) and the first inequality in (iii) of Lemma 6 follow immediately from representation (72) of the function 𝐺(𝑦,𝑡) and the next properties of the Wright functions: 0<𝑊−𝑧;−𝛽,𝛾≤𝐶exp−𝜅𝑧^{(1/(1−𝛽))}if𝛾,𝑧≥0, (A.1) where the constant 𝜅=𝜅(𝛽) will be positive if 𝛽∈(0,1) and either 𝛾≥1 or 0<𝛽≤𝛾<1. The left inequality (A.1) and the right one if 𝛾≥1 have been proved in Lemmas 2.2.4 and 2.2.6 in [21], where 𝐶=1/Γ(𝛾)<1 and 𝜅(𝛽)=𝛽^{(𝛽/(1−𝛽))}(1−𝛽).

Let us get the right estimate in the case of 0<𝛽≤𝛾<1. To this end, we use the well-known Wright formula (see e.g., (2.2.5) in [21] or [38]) and represent the function 𝑊(−𝑧;−𝛽,𝛾) as 𝑊.−𝑧;−𝛽,𝛾=𝛾𝑊−𝑧;−𝛽,𝛾+1+𝛽𝑧𝑊−𝑧;−𝛽,1+𝛾−𝛽 (A.2) As we consider the case of 𝛾−𝛽≥0 and 𝛾>0, which means 1+𝛾>1,1+𝛾−𝛽≥1, (A.3) we can apply inequality (A.1) to the terms in the right hand side of (A.2) and get 𝑊−𝑧;−𝛽,𝛾≤exp−𝛽^{(𝛽/(1−𝛽))}1−𝛽𝑧^{(1/(1−𝛽))}Γ𝛾+𝛽𝑧exp−𝛽^{(𝛽/(1−𝛽))}1−𝛽𝑧^{(1/(1−𝛽))}Γ.1+𝛾−𝛽 (A.4) It is easy to check that 1/Γ(𝛾), 𝛽/Γ(1+𝛾−𝛽)≤1 if 0<𝛽≤𝛾<1 and that 𝑧exp−𝛽^{(𝛽/(1−𝛽))}1−𝛽𝑧^{(1/(1−𝛽))}≤exp−𝛽^{(𝛽/(1−𝛽))}1−𝛽𝑧^{(1/(1−𝛽))}−𝛽,0≤𝑧≤1;exp^{(𝛽/(1−𝛽))}1−𝛽2𝑧^{(1/(1−𝛽))},𝑧>1. (A.5) Thus, the last inequalities together with (A.4) allow us to get (A.1) with 𝐶=2 and 𝜅=𝛽^{(𝛽/(1−𝛽))}(1−𝛽)/2>0 if 0<𝛽≤𝛾<1. Note that to prove (i) and the first inequality in (iii) of Lemma 6, we need (A.1) in the case of 0<𝛽≤𝛾<1.

To prove equality (36), we use formula (2.2.5) from [21] as follows: 𝑑^𝑛𝑑𝑡^𝑛𝑡^𝛾−1𝑊𝑐𝑡^−𝛽;−𝛽,𝛾=𝑡^{𝛾−1−𝑛}𝑊𝑐𝑡^−𝛽;−𝛽,𝛾−𝑛, (A.6) and represent the function 𝐺(𝑦,𝑡) as 𝐺()=𝜕𝑥,𝑡𝑊𝜕𝑡−𝑥𝑡^−𝛼/2𝑎₀^−1/2𝛼;−2,1. (A.7) After that, equality (A.7) and estimates (A.1) lead to ₀^+∞𝐺()𝛼𝑥,𝑡𝑑𝑡=𝑊0;−2,1. (A.8) Then, using definition (34) of the function 𝑊(𝑧;−𝛽,𝛾), we can deduce equality (36). To get inequalities (37), we need the well-known formula to a differentiation of the Wright function (see, e.g. (2.2.22) in [21] or [38]) as follows: 𝑑𝑊.𝑑𝑧𝑧;−𝛽,𝛾=𝑊𝑧;−𝛽,𝛾−𝛽 (A.9) Due to this formula and (A.6), we can represent 𝜕𝐺/𝜕𝑥,𝜕𝐺/𝜕𝑡 as 𝜕𝐺𝑡𝜕𝑥=−^{−1−𝛼/2}𝑎₀^1/2𝑊−𝑥𝑡^−𝛼/2𝑎₀^1/2𝛼;−2𝛼,−2;𝜕𝐺𝜕𝑡=𝑡⁻²𝑊−𝑥𝑡^−𝛼/2𝑎₀^1/2𝛼;−2,−1. (A.10) Furthermore, using inequalities (16) and (17) from Lemma 3 [22], we infer the estimate of the Wright function 𝑊−𝑧;−𝛽,𝛾≤const.1+|𝑧|^{−((𝛾−1)/𝛽)},if𝛾<1. (A.11) Thus, representations (A.10) and inequality (A.11) lead to (37). Note that statements (iv)–(vii) of Lemma 6 follow, after some simple calculations, from representations (A.10) and estimate (A.11).

In virtue of formulas (2.2.4) in [21] or (11) and (14) in [41]: 𝐷^𝜈_𝑡𝑡^𝛾−1𝑊−𝑐𝑡^−𝛽;−𝛽,𝛾=𝑡^{𝛾−𝜈−1}𝑊−𝑐𝑡^−𝛽,𝐼;−𝛽,𝛾−𝜈_𝜈𝑡𝑡^𝛾−1𝑊−𝑐𝑡^−𝛽;−𝛽,𝛾=𝑡^{𝛾+𝜈−1}𝑊−𝑐𝑡^−𝛽;−𝛽,𝛾+𝜈,ifRe𝜈>0, (A.12)

and equality (A.9), we state (42).

B. The proof of Lemma 10

Using estimate of Lemma 3.1 in [26] or Theorem 1.6 in [29]: 𝐸_𝛾,𝛽(𝑧≤𝐶)1+|𝑧|,𝜇≤arg𝑧≤𝜋, (B.1) we have got 𝐸_𝛼/2,𝛼−𝑏₀𝑎₀^−1/2𝑡^𝛼/2≤𝐶1+𝑡^𝛼/2,𝜇=2𝜋𝛼3. (B.2) Then 𝐺₂(𝑡)≤𝐶𝑡^𝛼−11+𝑡^𝛼/2. (B.3) Due to formula (E.32) in [42], we can conclude that 𝐺₂(𝑡) is positive for 𝑡>0. Thus, the statement (i) of Lemma 10 follows immediately from (B.3) and the arguments above.

Due to formula (1.82) in [29]: 𝐼^𝜈_𝑡𝑡^𝛽−1𝐸_𝛾,𝛽𝜆𝑡^𝛾=𝑡^{𝛽+𝜈−1}𝐸_{𝛾,𝛽+𝜈}𝜆𝑡^𝛾𝐷,Re𝜈>0,_𝜈𝑡𝑡^𝛽−1𝐸_𝛾,𝛽𝜆𝑡^𝛾=𝑡^{𝛽−𝜈−1}𝐸_{𝛾,𝛽−𝜈}𝜆𝑡^𝛾, (B.4)(B.5)

we can calculate value of 𝐼_𝑡^1−𝛼𝐺₂(𝑡) and get 𝐼_𝑡^1−𝛼𝐺₂(𝑡)=𝐸_𝛼/2,1−𝑏₀𝑎₀^−1/2𝑡^𝛼/2=^+∞_𝑘=0−𝑏₀𝑎₀^−1/2𝑡^𝛼/2𝑘Γ((.1+𝛼𝑘/2)) (B.6) Here we used the definition of 𝐸_𝛾,𝛽(𝑧) (see (89)). Then, representation (B.6) leads to (91). Note that 𝐷^𝛼_𝑡𝐺₂(𝑡)=(𝑑/𝑑𝑡)𝐼_𝑡^1−𝛼𝐺₂(𝑡). Hence ₀^+∞𝐷^𝛼_𝜏𝐺₂(𝜏)=𝑑𝜏_0+∞𝑑𝐼𝑑𝜏^𝜏1−𝛼𝐺₂(𝜏)𝑑𝜏=lim_𝜏→+∞𝐼_𝜏^1−𝛼𝐺₂(𝜏)−lim_𝜏→0𝐼_𝜏^1−𝛼𝐺₂(𝜏), (B.7) or due to (B.6) and (91), ₀^+∞𝐷^𝛼_𝜏𝐺₂(𝜏)𝑑𝜏=lim_𝜏→+∞𝐸_𝛼/2,1−𝑏₀𝑎₀^−1/2𝜏^𝛼/2−1. (B.8)

To obtain (92), we apply estimate like (B.1) to the first term in the right-hand side (B.8). In the same way, we infer inequality (95).

Inequalities (93)–(95) follow from (B.5)–(B.8) and (B.1).

At last, proving inequality (96) is enough to use formula (1.83) in [29] as follows: 𝑑𝑡𝑑𝑡^𝛽−1𝐸_𝛾,𝛽𝑡^𝛾=𝑡^𝛽−2𝐸_{𝛾,𝛽−1}𝑡^𝛾, (B.9) and to estimate (B.1).

Acknowledgments

The authors are very grateful to the referee for the careful reading of the paper and the valuable comments. This work is partially supported by Grant PIRSES-GA-2011-295164.