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

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 initialboundary value problems for the equation ∂α t u(x, t) = Lu(x, t) + f 0 (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.

Concerning problems (1)-( 6), they have the following features.First of all, (4) is a fractional dynamic boundary International Journal of Partial Differential Equations condition; next, these problems are formulated for the subdiffusion linear equation.
Note that if  = 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][3][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  = 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][6][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][11][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 ⟩ ∼   .If  ∈ (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   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 -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 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  +  .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 A = T(), where T() is a nonlinear function of  and A is the linear operator derived in Section 4; that is, A −1 T is the solution of the model problem for data T. Setting S = A −1 T, we will then prove that the mapping  → S(), where S() = A −1 T, 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.

Model Problems
Let  + = (0,+∞), and  +  =  + × (0, ), and  0 and  0 be some positive numbers.Here we will discuss the first initial-boundary value problem for the fractional diffusion equation in  +  and the initial-boundary value problem with the fractional boundary condition in  +  .
Denote by ŵ(, ) the Laplace transformation of the function (, ); that is, The Laplace transformation in ( 22)-( 24) leads to the problem Here we used the following formula from [33]: One can easily check that the following function solves the equations in (29): Due to formula (2.30) in [34] and the inverse Laplace transformation, we get the integral representations of V(, ) as follows: where Here (; , ) is the Wright function, which is defined for , ,  ∈ C as (see formula (1.8.1( 27)) in v.3 [35]) 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].
To complete the proof of Lemma 7, we need to obtain inequality (45).Let  1 ,  2 ∈ [0, ] and  2 >  1 .Denote We analyze the difference As for the last term in this sum, it is estimated by . We change the variable  =  1 −  in the term J 1 and apply estimate (38) with  := Δ,  := .Thus, we have In the same way, we evaluate the function J 2 .The estimate of the term J 3 follows from the properties of the function     1 and inequality (36).At last, the mean-value theorem together with estimate (40), where  := Δ,  := , lead to 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(, ) ∈  2+,((2+)/2) ( +  ) of problem ( 22)- (24), which is represented with (32) and Proof.First of all we obtain estimate (61).One can get the following inequality using the results of Lemma 7 and ( 22), where  0 ≡ 0 : Next, we use formula (3.5.4) from [1] as follows: to evaluate the maximum of |V(, )|.Hence, (43) and (63) lead to inequality sup here we use the fact that V(, 0) = 0.
After the application of the Laplace transformation in time  to problem (82), we have the following: Here, we used again formula (30).Some simple calculations lead to the function 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 where the kernel (, ) is given by (33) and Here,  , () 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]) Note that this two-parameter function of the Mittag-Leffler type was in fact introduced by Agarwal [40].
The function (, ) has been studied in Section 3.1 (see Lemma 6).Thus, to describe the properties of the function (, ), we have to observe the function ().To this end, we will use the following properties of the kernel  2 (), which are proved in Appendix B.
Lemma 10.Let , , and  be some positive constants,  ∈ (0, 1);  ]  and  ]  be the fractional Riemann-Liouville integral and derivative, correspondingly (their definitions are given in (49) and (47)).Then the following is true.belongs to  ((3+)/2) ([0, ]), (0) = 0, and the following estimate holds: Proof.First of all, we evaluate the value of max ( This inequality gives that Next, we obtain the representation of    ().Due to equality (99) and properties ( 46) and (48), we conclude that Since Φ 1 (0) = 0, we can rewrite the last equality as or, applying (91) and (92), we have Changing the variable  2 −  =  in B 1 and  1 −  =  in B 2 , we get Then, the property of the function Φ Proof.To prove this theorem in the case of (84) it is enough to consider the following Dirichlet problem: and to apply the results of Theorem 9 and Lemma 12.

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  2+,((2+)/2) (Ω  ) of the next initial-boundary value problem: (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.
Proof.If the right-hand sides of (1)-( 5) meet the following requirements: 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 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 (, ) ∈  2+,((2+)/2) (Ω  ).The next step deals with the proving of the following equalities: We obtain after simple calculations the following properties for the functions   (), for all  ∈ [0, ]: After that, using (129), one can easily check that the right-hand sides in (126) satisfy conditions of Theorem 14.

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: We require the following conditions on the functions   ,    , ,  = 0, 1;  = 1, 2: (i) There exist some positive constants   ,  = 0, 7 such that 0 () ∈  2+ (Ω) .
We introduce the functional spaces H and H 0 as follows: and for any elements  = (V, Let   (V) = {V ∈ H : ‖V‖ H ≤ } and  0  (V) = {V ∈ H : ‖V‖ H ≤ , V(, 0) = 0},  <  0 , be balls of radius  in the space H, centered at the origin, for some positive  0 to be determined later on.
The simple calculations lead to the following assertion.
First, we linearize problem (148) on the initial data and represent one as a system A = T(), where A is a linear operator and T() is a nonlinear perturbation.To this end, we introduce the new unknown function as follows: where the function  0 (, ) is a solution of the following problem: By Theorem 5 and Proposition 16, there exists a unique solution  0 (, ) of the problem (159) and where the positive constant  0 depends on   ,  = 0, 7,   ,   , ,  = 0, 3, .
Next, we rewrite problem (148) in terms of the function (, ) and after some calculations get the problem in the form: −  0 ( 0 ,  0 )   = Φ 0 (,   ,   ) , (, ) ∈ Ω  ,  ∈ (0, 1) ;  (, 0) = 0 in Ω; where Thus, we represent nonlinear problem (148) as Note that if we froze the functional arguments in the functions Φ 0 (,   ,   ) and Φ  1 (,   ),  = 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 and S is a nonlinear operator.We will show that S is a contraction operator.
Next, we will obtain inequality (169).Note that, as it follows from definitions (162) and ( 164 (178) After that, the simple calculations together with the results of Proposition 16 allow us to get the following for any function (, ) ∈  0  (). ( It is obviously  2 (, ) → 0 as ,  → 0.
Using the analogous arguments, it is possible to assert the following results.