On time fractional pseudo-parabolic equations with nonlocal integral conditions

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order \begin{document}$ \sigma,\; \; 0 and the space fractional derivative is of order \begin{document}$ \alpha,\beta >0 $\end{document} . In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen \begin{document}$ \alpha, \beta $\end{document} . The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in \begin{document}$ L^p $\end{document} between the regularized solution and the sought solution is obtained.

1. Introduction. Fractional differential equations (FDEs) have been extensively studied during the past two decades by many researchers because of their diverse applications in physics, electrochemistry, viscoelasticity, etc. Time-fractional PDEs have been used as a major tool for modeling various practical fields and there are a number of publications devoted to the study of time-fractional PDEs and their applications (we refer the reader to [2,28,18,20]).
In this paper, for α, β ∈ (0, 1), we consider the integral boundary problem for the fractional differential equation as follows in Ω × (0, T ], in Ω. (P) Here we consider a domain Ω ⊂ R N with the smooth boundary ∂Ω and the constants ρ 1 , ρ 2 ≥ 0 satisfying ρ 2 1 + ρ 2 2 > 0. Our problem is studied with the time fractional derivative of order σ ∈ (0, 1) in the sense of Caputo which is denoted by ∂ σ t . In the main equation of problem (P), if we take κ = 0, and σ = α = β = 1, we have the usual parabolic equation, which has been investigated by many researchers; see [15], [23], [19] and the references therein. If σ = α = β = 1, and κ > 0, our main equation becomes the pseudo-parabolic equation. This type of FDEs can be used to model many phenomena in many fields of science. In [8], Peter J. Chen and Morton E. Gurtin presented a theory about a non-simple material for which the conductive temperature and the thermodynamic temperature do not coincide. The nonstationary processes in semiconductors in the presence of sources can be analyzed by the equation (see [31]) The unidirectional propagation of nonlinear, dispersive, long waves is also described by the classical pseudo-parabolic (see [4]). For more applications, we refer the reader to [9], [24]. Problems with the usual Cauchy conditions such as the initial condition u(x, 0) = φ(x) or the condition at the terminal time t = T are familiar. Usually, we can obtain well-posedness results for problems with initial conditions. In contrast, problems with Cauchy conditions at the terminal time are often ill-posed; we refer the reader to some recent results [17,25] on the terminal value problem. Our paper considers a non-local in time condition replacing the usual Cauchy conditions, that is In [12], [30], we can find two types of condition similar to the above such as M. Beshtokov [7,5,6] considered boundary value problem for FPPDE. In practice, some phenomena will be simulated more effectively if we investigate the problems with the non-local condition. Indeed, in some models for meteorology, the timeaveraged data help us to get a more reliable long-term weather forecast (see [3]). The problem with this type of condition can also be used when investigating radionuclides propagation in Stokes fluid, diffusion and flow in porous media ( [13], [22], [26]). Compared with usual local initial/final value conditions, non-local conditions are more difficult to handle and motivated by this reason and their high application value, we work on time-fractional pseudo-parabolic equations with non-local final conditions, and our paper provides new results for the linear source term case (to the best of the authors' knowledge, it seems that a problem like (P) has not really been studied). Our paper will investigate problem (P) and the main results of this work are as follows: • The regularity results for the mild solution.
• The proof for the instability of the solution to the initial data recovery problem.
• The regularization of the initial data recovery problem. This paper is organized as follows. Section 2 gives some preliminaries that are needed throughout the paper. In section 3, we give the regularity result for the mild solution, and an example which shows that our solution is unstable in the case t = 0, and moreover, we give a regularized solution for the initial data recovery problem.

2.
Preliminary. Before we introduce the main results of our works, some preliminary materials are given.
Definition 2.1. Let · B be a norm on a Banach space B. Then, we define the following spaces (see [16]) The fractional Laplace operator of order α ∈ (0, 1) is defined as a Fourier multiplier with symbol −|ξ| 2α given by and it is equivalent to where the notations F and F −1 stand for the Fourier transform and the inverse Fourier transform, respectively.
Next, let us recall the spectral problem for the fractional Laplace operator on the bounded domain Ω as follows where the sequence of positive eigenvalues {λ j } j∈N satisfy whose corresponding set of real eigenfunctions {ψ j (x)} j∈N is orthogonal and complete.
Remark 2. If the inner product on L 2 (Ω) is denoted by (·, ·) L 2 (Ω) , then the Fourier series of a function u in L 2 (Ω) can be formulated as Definition 2.4. For any η > 0, we define the fractional Hilbert scale space by for w ∈ H −η where (·, ·) * represents the dual product between H η (Ω) and H −η (Ω). (Ω), respectively. Then we have For more details about the definition of the fractional Sobolev spaces, we refer the reader to [10].
Definition 2.6. For α > 0, and a arbitrary constant β ∈ R, the Mittag-Leffler function can be defined by (see [14]) where Γ is the usual Gamma function.
Lemma 2.7. (see [14])For 0 < α 1 < α 2 < 1 and α ∈ [α 1 , α 2 ], there exist positive constants m α and M α , depending only on α such that 3. Main results. Using the Laplace transform method, we can find the formula of the solution to the first equation of (P) as follows To find the formula of the mild solution to problem (P), we need to find the representation of the initial data u(x, 0). Using our non-local final condition, we have Therefore, the formula of the mild solution to the problem (P) can be given by Next we introduce the structure for this section. Part 1: Regularity of the mild solution. Part 2: The ill-posedness of the initial data recovery problem.
Part 3: Regularization and L p error estimate for the initial data recovery problem.
3.1. Regularity result. Lemma 3.1. Let 0 < α, β < 1. Then we can find a constant C 0 > 0 such that (ii) If α 2 < β < 1, we have Proof. First, we use the Cauchy inequality to get Using Lemma 2.7, it follows that From the above estimate, we find that • If α 2 < β < 1, we have From the above estimates, our lemma is proved.
Theorem 3.2. We assume that the constants σ, α, β, θ, η, p, k, m satisfy Then the mild solution u of the Problem (P) will belong to L p (0, T ; W k,m (Ω)) and the following holds u L p (0,T ;W k,m (Ω)) f H η+ α 2 (Ω) + F L ∞ (0,T,H η−βθ+ α 2 (Ω)) where we use the notation a b if we can find a positive constant K such that a ≤ Kb.
Proof. By the triangle inequality, the following holds Hence, we need to estimate the four terms on the right-hand side of the above to obtain the regularity results for our mild solution.
• Estimate of the first term. Parseval's identity gives us Thanks to Lemma 3.1 and Lemma 2.7, we obtain The Sobolev embedding H η+ α 2 (Ω) → H η− α 2 (Ω) enables us to get • Estimate of the second term.
Using Lemma 2.7 we have Now, the Hölder inequality will be applied to get the following estimate Combining these and noting that λ 2η−2βθ−α j λ 2η−2βθ+α j , ∀j ∈ N, we deduce that In the same way as in the previous step, we obtain Note that, we get the estimate above by using the Hölder inequality as follows We thus have • Estimate of the fourth term. This term can be treated more simply than the above two terms. Indeed, thanks to Lemma 2.7 and the Hölder inequality, we have . Combining these estimates with (17), we have We apply the following Sobolev embeddings Let us note that the integral can assert that This show that u belongs to L p (0, T ; W k,m (Ω)) and the proof is complete.

3.2.
The ill-posedness of the initial data recovery problem. From now on, we will only consider our problem in the homogeneous case i.e. when F = 0. Furthermore, we also assume that β > α throughout the rest of the paper.
Lemma 3.3. Let σ ∈ (0, 1) and 0 < α < β < 1. Then, we get the following estimate Proof. Lemma 2.7 gives us In the latter inequality, we note that the integral Theorem 3.4. If t = 0, the solution of the problem (P) is unstable in the sense of the L 2 (Ω) norm.
Proof. We begin the proof by setting u(x, 0) = u 0 (x) and defining a mapping T from L 2 (Ω) to L 2 (Ω) as follows where It's a simple matter to check that ϕ(x, z) = ϕ(z, x), and T is a self-adjoint operator. Let us consider the following finite rank operator From Lemma 3.3, it is clear that It follows immediately that T − T M L (L 2 (Ω);L 2 (Ω)) −→ 0 as M → ∞. We also can prove that T is compact. Moreover, we have and then, we can conclude that our problem is ill-posed. Let us give an example to illustrate the ill-posedness of the problem. Taking the input data f k (x) = 1 λ β k + 1 λ β−α k ψ j (x), we can see at once that However, the initial data u 0,k corresponding to the final data f k , is given by and it follows that The above estimate helps us to get the limit below From (39) and (42), we deduce that the solution to problem (P) is unstable in L 2 (Ω).

3.3.
Regularization and L p error estimate.
Theorem 3.5. Let f ε be noisy data satisfying f ε − f L p (Ω) ≤ ε, for p ≥ 1, and u 0 ∈ H ν (Ω) for ν > 0. Then, we can find a regularized solution u 0,ε such that Proof. First, for θ ∈ (0, 1), we set M ε = ε θ−1 γ+β . Then, we choose the following regularized solution and define the supporting series as follows For the purpose of obtaining an error estimate between u 0,ε and u 0 , we need two important estimates below.
• For any γ < min N 2 , ν , thanks to Lemma 3.1, we have (48) • From the formula of u 0 , we deduce that It is easily seen that when ε → 0, we have and this finishes our proof.

4.
Conclusion. This paper investigates time fractional pseudo-parabolic equations with nonlocal integral conditions. The results of this work are divided into two main parts: Part I: The Regularity result of the mild solution for Problem (P) is given. Part II: We show that the initial data recovery problem is ill-posed. We also give the regularized solution and estimate error in L p between the regularized solution and the sought solution. This paper only considers the linear case, and in the future we hope to expand Problem (P) to the case of nonlinear source terms.