UNIQUE SOLVABILITY OF A DIRICHLET PROBLEM FOR A FRACTIONAL PARABOLIC EQUATION USING ENERGY-INEQUALITY METHOD

In this paper, we establish su cient conditions for the existence and uniqueness of the solution in fractional functional space for a class of initial boundaryvalue problems for a class of partial fractional parabolic di erential equations that include a fractional derivative of Caputo. The results are established by the application of the method based on a priori estimate "energy inequality" and the density of the range of the operator generated by the problem considered. áâ ­®¢«¥­÷ ¤®áâ â­÷ ã¬®¢ ̈ ÷á­ã¢ ­­ï â  õ¤ ̈­®áâ÷ à®§¢'ï§aã § ¤à®¡®¢®£® äã­aæ÷®­ «ì­®£®  ̄à®áâ®àã ¤«ï ®¤­®£® a« áã  ̄®ç âa®¢®-aà ©®¢ ̈å § ¤ ç ¤«ï ¤¥ïa ̈å ¤à®¡®¢®̄ à ¡®«÷ç­ ̈å ¤ ̈ä¥à¥­æ÷ «ì­ ̈å à÷¢­ï­ì ÷§ ¤à®¡®¢®î  ̄®å÷¤­®î   ̄ãâ®. ¥§ã«ìâ â ̈ ®âà ̈¬ ­® è«ïå®¬ § áâ®áã¢ ­­ï ¬¥â®¤ã ¥­¥à£¥â ̈ç­ ̈å ­¥à÷¢­®áâ¥©. ®¢¥¤¥­  é÷«ì­÷áâì ®¡à §ã ® ̄¥à â®à , é® ¢÷¤ ̄®¢÷¤ õ § ¤ ç÷.


Introduction
Fractional dierential equations (FDEs) are obtained by generalizing dierential equations to an arbitrary order. Since fractional dierential equations are used to model complex phenomena, they play a crucial role in engineering, physics and applied mathematics. Therefore they have been generating increasing interest from engineers and scientist in recent years. Since FDEs have memory, nonlocal relations in space and time, complex phenomena can be modeled by using these equations. Due to this fact, materials with memory and hereditary eects, through strongly anomalous media. Indeed, we can nd numerous applications in viscoelasticity, electro-chemistry, signal processing, control theory, porous media, uid ow, rheology, diusive transport, electrical networks, electromagnetic theory and probability, signal processing, and many other physical processes are diverse applications of FDEs [1 -7].
The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional dierential equations have been studied extensively in a large cycle works. But there are not many works in the fractional eld of partial dierential equation, this is due to the diculty of applying classical theories and methods to a eld of farctional partial dierential eqation. Motivated by this, we conducted a detailed and thorough study in this eld to see the behavior of the solution to these problems using the classic energy estimat method. Then, the present Keywords. Partial fractional dierential equation, energy inequality; existence; uniqueness.

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paper is devoted to the study of initial-boundary value problem for a parabolic equation with time-fractional derivative with Dirichlet condtion, which has not been studied so far.

Preliminaries and functional spaces
Let \Omega = [0, T ] be a nite interval of the real numbers \BbbR and \bfGamma (\cdot ) denote the gamma function. For any positive integer 0 < \alpha < 1, the Caputo derivative are the Riemann Liouville derivative are, respectively, dened as follows:Let \Gamma (.) denote the gamma function. For any positive integer 0 < \alpha < 1, the Caputo derivative are the Riemann Liouville derivative are, respectively, dened as follows: (1) The left Caputo derivatives: (2.1) (2) The left Riemann-Liouville derivatives: (4) The right Riemann-Liouville derivatives: Many authors think that the Caputo's version is more natural because it allows the handling of inhomogeneous initial conditions in a easier way. Then the two denitions (2.1) and (2.2) are linked by the following relationship, which can be veried by a direct calculation: (2.3) Denition 2.1. [37] For any real \sigma > 0 and nite interval [a, b] of the real axis Rÿ we dene the semi-norm: and norm: we then dene l H \sigma 0 (\Omega ) as the closure of C \infty 0 (\Omega ) with respect to the norm \| \cdot \| l H \sigma (\Omega ) .

Statement of problem
In the rectangular domain Q = (0, d) \times (0, T ), with d, T < \infty and 0 < \alpha < 1, we shall study the existence and uniqueness of solutions u = u(x, t) to the following fractional parabolic problem : We consider the following fractional parabolic equation of the type: with the initial condition : and Dirichlet condition : Where a, b, \widetil f and \varphi are known functions. We shall assume that the function \varphi satises a compatibility conditions, i.e., And the functions a, b verify: with the initial condition

Estimation a priori
The method used here is one of the most ecient functional analysis methods and important techniques for solving partial dierential equations with integral conditions, which has been successfully used in investigating the existence, uniqueness, and continuous dependence of the solutions of PDE's, the so-called a priori estimate method or the energy-inequality method. This method is essentially based on the construction of multiplicators for each specic given problem, which provides the a priori estimate from which it is possible to establish the solvability of the posed problem. More precisely, the proof is based on an energy inequality and the density of the range of the operator generated by the abstract formulation of the stated problem, so to investigated the posed problem, we introduce the needed function spaces. In this paper, we prove the existence and the uniqueness for solution of the problem   Proof. Multiplying the equation (3.1) by the following function: and integrating over Q = (0, d) \times (0, T ), we get As .

By integration by parts over
By using the Cauchy inequality with \varepsi , for \varepsi < 2b 0 ; and becaude of the equivalent of the semi-norms | .| l H \sigma (Q) and | .| c H \sigma (Q) , there is a positive constant m such that

So, we have
which gives that Hence, we get So, we obtain As all the temes are positive, we have Finally, it follows that Therefore, we obtain that Hence the uniqueness of the solution.
\square Remark 4.2. This inequality\| v\| B \leq k \| Lv\| F is gives the uniqueness of the solution, indeed.
Let v 1 and v 2 two solutions, so which gives the uniqueness of the solution.
Proposition 4.3. The operator L from B to F admits a closure.
The convergence of v n toward 0 in B entails that v n \rightar 0 in (C \infty 0 (Q)) \prime .
(4.15) On the other hand the convergence of \scrL v n to f in F = L 2 (Q) implies that \scrL u n \rightar f in (C \infty 0 (Q)) \prime . (4.16) By virtue of the uniqueness of the limit in (C \infty 0 (Q)) \prime , we conclude from (4.9) and (4.10) that f \equiv 0.
Hence, the operator L is closable.   So there is a corresponding sequence (v n ) n \subset D(L) such that Lv n = z n .
We can deduce that (v n ) n is a Cauchy sequence in B, so there is v \in B: By virtue of the denition of L (\mathrm{ \mathrm{ \mathrm{ n\rightar+\infty v n = v in B ; if \mathrm{ \mathrm{ \mathrm{ n\rightar+\infty Lv n = \mathrm{ \mathrm{ \mathrm{ n\rightar+\infty z n = z, so \mathrm{ \mathrm{ \mathrm{ n\rightar+\infty Lv n = z and as L is closed so Lv = z), the function v verify that So we conclude here that R(L) is closed because it is complete (any complete subspace of a metric space (not necessarily complete) is closed).
It remains to show the opposite inclusion. Let z \in R(L), then there is a sequence of (z n ) n in F consists of the elements of the set R(L) such that \mathrm{ \mathrm{ \mathrm{ n\rightar+\infty z n = z.
where z \in R(L), because R(L) is closed subset of a complete space F , then R(L) is complete.
So there is a corresponding sequence (v n ) n \subset D(L) such that Lv n = z n .
From the estimate (4.11), we obtain \| v p -v q \| B \leq k \bigm\| \bigm\| Lv p -Lv q \bigm\| \bigm\| F \rightar 0, if p, q \rightar +\infty . We can deduce that (v n ) n is a Cauchy sequence in B, so there is v \in B: Once more, there is a corresponding sequence (L (v n )) n \in R(L) such that Lv n = Lv n over R(L), \forall n.

Existence of solution
To show the existance of solutions, we prove that R(L) is dense in F for all u \in B and for arbitrary \scrF = (f, 0) \in F. Proof. The idea of the proof of the theorem is choose w \in R(L) \bot (exactly w \in R(L) \bot \subset L 2 (Q)) and for all v \in B, we have and demonstrate that R(L) \bot = \{ 0\} which give The scalar product of F is dened by The equality (5.1) can be written as follows: where C D \alpha t v, \partialv \partialx , v \in L 2 (Q), with v satises the boundary conditions of (3.3). From Replacing w in (5.4) by its representation (5.5) and integrating by parts each term of (5.4) and by taking the condition of v, we obtain