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 initial-boundary value problems for the equation , , 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.
Let and be any numbers from and let , ; ; ; be a fixed value. In this paper, we consider a partial differential equation with the fractional derivative in time as follows: Here, denotes the Caputo fractional derivative with respect to and is defined by (see, e.g., in ), where is the gamma function, , , are the given functions, and is a positive. Note that if , then (1) represents a parabolic equation. As we are interested in the fractional cases, we restrict the order to the case .
We will solve (1) satisfying the following conditions: the fractional dynamic boundary condition on : and one of the following conditions on : the Dirichlet boundary condition: or the Neumann boundary condition: or the fractional dynamic boundary condition:
Here, and are given positive functions, and , , and , , are given functions.
Note that if , 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 , 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 . 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  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  in the Hölder spaces and by Bizhanova and Solonnikov  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 scales as a nonlinear power law in time, that is, . If , 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 , glass forming materials , and biological media . The review paper by Klafter et al.  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  solved a nonhomogeneous fractional diffusion-wave equation in a one-dimensional bounded domain. Metzler and Klafter , 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  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  and Sakamoto and Yamamoto  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 , 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.  which is an encyclopedic treatment of the fractional calculus and also Kilbas et al. , Mainardi , Podlubny , and Pskhu .
As for the quasilinear equation like (1), Clément et al.  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  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  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 , where is a nonlinear function of and is the linear operator derived in Section 4; that is, is the solution of the model problem for data . Setting , we will then prove that the mapping , where , 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 be a bounded or an unbounded domain in , , ; ; . Denote
Definition 1. We will say that functions and that if and only if the functions and , , are continuous and the following norms are finite:
Note that if , the spaces , , coincide with the ordinary Hölder spaces (see (1.10)–(1.12) in Chapter 1 in ). Further, we also use the Banach spaces and of the functions and with the finite norms In a similar way, we introduce the spaces , .
Definition 2. We will say that functions and that if and only if and and and .
If ,, and , then .
We introduce the spaces of , , with the same way.
Let and be some positive numbers. We assume that the given functions , , and , , in (1), (4), and (7) are subject to the following conditions: and one of the following: or or Note that requirements (15)–(18) are called the consistency conditions.
The main results of our paper are the following:
Theorem 3. Let , and conditions (13)–(16) hold, and , , , for any positive number . Then there exists a unique solution of problem (1)–(5): , , and where a positive constant depends only on the measure of and , , , .
Theorem 4. Let , and conditions (13)–(15), and (17) hold, and let , , , for any positive number . Then there exists a unique solution of problem (1)–(4), (6) as follows: , , and where is a positive constant and depends only on , , , , and the measure of .
Theorem 5. Let , and conditions (13)–(15) and (18) hold, and , , , and for any positive number . Then there exists a unique solution of problem (1)–(4), (7) as follows: , , , and where is a positive constant and depends only on ,, , , and the measure of .
3. Model Problems
Let , and , and and 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 .
3.1. The Solvability of the First Initial-Boundary Value Problem for the Subdiffusion Equation
We look for the function by the following conditions: where and , are some given functions.
We assume that the following conditions hold: for some positive number .
Note that conditions (25) and (26) together with restriction (27) allow us to apply the Laplace transformation in to the right hand sides of (22)–(24). Indeed, conditions (25)–(27) mean that the right hand sides in (22)–(24) except equal zero and . Thus, we can extend the right hand sides in (22)–(24) by for and save, for simplicity, the same notation for the extension of the function . Therefore, we can apply, at least formally, the Laplace transformation in to (22)–(24) in the case of (25)–(27) hold.
Denote by 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 : One can easily check that the following function solves the equations in (29): Due to formula (2.30) in  and the inverse Laplace transformation, we get the integral representations of as follows: where Here is the Wright function, which is defined for as (see formula () in v.3 ) The main properties of the Wright functions are described in Chapters 4.1, v.1 and 18.1 v3 in , Chapter 1.11 in , Chapter 1.3 in , and Chapter 2 in [21, 36].
Lemma 6. Let , , and be some positive constants, , , , . Then one has the following. where is the Riemann-Liouville fractional derivative, and its definition is in (2.1.8) in .
Proof. First, we obtain the representation of . To this end, we need the following properties of the fractional derivative (see Lemma 2.10 and formula () in ): (i)
where (see () in )
(ii) If the functions , , and are bounded in , then
where is the Riemann-Liouville fractional integral of order (see, e.g. () in )
One can easily see that and (see (25) and (27)). Then, using properties (46) and (48) and equality (36), we represent the function as
Namely, this representation will be useful below. To prove inequality (43), we will use this representation, statement from Lemma 6 and the first estimate in (37) as follows: In view of , if , we can rewrite representation (50) as
Let , and . Denote Then,
Using inequality (38) with , , we obtain As for the term , we apply the mean-value theorem to the difference together with inequality (39) (where , ) and deduce 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 and . Denote We analyze the difference As for the last term in this sum, it is estimated by . We change the variable in the term and apply estimate (38) with , . Thus, we have In the same way, we evaluate the function . The estimate of the term follows from the properties of the function and inequality (36). At last, the mean-value theorem together with estimate (40), where , , lead to Therefore, inequality (45) is deduced from (58)–(60).
Proof. First of all we obtain estimate (61). One can get the following inequality using the results of Lemma 7 and (22), where
Next, we use formula () from  as follows:
to evaluate the maximum of . Hence, (43) and (63) lead to inequality
here we use the fact that .
After that, the minor seminorms of the function are estimated with the interpolation inequalities from Section 8.8  and (43)–(45), (62), and (64). Therefore, the arguments above prove inequality (61) and the embedding .
Next, we show that the function given by (32) satisfies (22). To this end, we use equality (46) and (48) and represent as Then (41) leads to Next, due to property (42) of the function , one can check that As it follows from the first inequality in (37), the function represented by (32) satisfies the following conditions:
Finally, it is necessary to show that the function meets boundary condition (24). To this end, we observe the next difference (here we will essentially use statement (35)) Applying inequality (38) with , to the term , we get To estimate the term in (69), we use the Wright formula (see  or () in ): and rewrite the function as After that, due to Lemmas and in  and representation (72), we have Thus, Then, we joint estimates (69)–(74) and obtain which means that .
Therefore, as it was written above, the function 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 .
Now we remove restriction (27). To this end, it is enough to consider the Cauchy problem: Here, and are extensions of the functions and , correspondingly, onto . These functions together with their corresponding derivatives have finite supports and The results of Theorem 3.2 from  give the one-valued solvability of (76) and or, due to inequalities (77), After that, we will look for the solution of problem (22)–(24) as where 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 . This fact and the properties (see (77)) of the function allow us to obtain the next results.
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 by the following conditions: where are the given functions. Let the following conditions hold: for some positive number .
At the beginning, we assume that and search a solution of (82) under this restriction in the class , if , .
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  and (1.80) in , 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  or () in ) Note that this two-parameter function of the Mittag-Leffler type was in fact introduced by Agarwal .
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 , which are proved in Appendix B.
Lemma 10. Let , , and be some positive constants, ; and be the fractional Riemann-Liouville integral and derivative, correspondingly (their definitions are given in (49) and (47)). Then the following is true.
Proof. First of all, we evaluate the value of . To this end, we use the first inequality in (90). Thus, one has This inequality gives that Next, we obtain the representation of . Due to equality (99) and properties (46) and (48), we conclude that Since , we can rewrite the last equality as or, applying (91) and (92), we have To estimate , we use representation (102) and get