Final Value Problem for Parabolic Equation with Fractional Laplacian and Kirchhoff’s Term

In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when δ tends to zero.


Introduction
The aim of this study is to investigate the final value for the space fractional diffusion equation where the symbol C ∂ α vðtÞ/∂t α is called the conformable derivative which is defined clearly in Section 2. Here, Ω ⊂ ℝ d ðd ≥ 1Þ is a bounded domain with the smooth boundary ∂Ω, and T > 0 is a given positive number. The function F represents the external forces or the advection term of a diffusion phenomenon, etc., and the function f is the final datum which will be specified later. The applications of the conformable derivative are interested in various models such as the harmonic oscillator, the damped oscillator, and the forced oscillator (see, e.g., [1]), electrical circuits (see, e.g., [2]), chaotic systems in dynamics (see, e.g., [3]), and quantum mechanics (see, e.g., [4]). From the paper, see, e.g., [5], we must confirm that the study of the ODE problem with the conformable derivative is very different from the study of the PDE problem with a conformable derivative. Results and research methods of the wellposedness for the ODE and PDE model are not the same and are completely different. The following two remarks confirm what we have just pointed out. Remark 1. Let us first discuss conformable ODEs. Let v be the functions whose domain of its value is ℝ. If α = 1, C ∂ α /∂t α becomes the classical derivative. If 0 < α < 1, by the paper of [6], we know that the relation between the conformable derivative and the classical derivative by the following lemma.

Lemma 2.
If v : ½0, T ⟶ ℝ, then a conformable derivative of order α at s > 0 of v exists if and only if it is differentiable at s, and the following equality is true: Remark 3. In the following, we mention the PDEs with conformable derivative where D is a Sobolev space, such as L 2 ðΩÞ, W γ,p ðΩÞ, and DðA γ Þ. When we study the PDE model, we often do with a multivariable function v : ð0, TÞ ⟶ D, where D is a Sobolev space. This means that, for each t, vðtÞ can take on values in many classes of spaces with D 1 ↪D↪D 2 ⋯ . Some illustrated examples given in [5] say that (2) may be not true on Sobolev spaces.
When α = 1, the main equation of Problem (1) appears in many population dynamics. By the work of Chipot and Lovat [25], we know that the diffusion coefficient B is dependent on the entire population in the domain instead of local density; that is, the moves are guided by considering the global state of the vehicle. The function u is a descriptive population density (e.g., bacteria) spread. According to article [26], we find that model (1) is a type of Kirchhoff equation, arising in vibration theory; see, for example, [27].
(i) This paper is the first study on the final value problem for a diffusion equation with a Kirchhoff-type equation and conformable derivative. Since our models are nonlinear, in order to establish the existence and uniqueness of solutions, we have to use the Banach contracting mapping theorem combined with some techniques to evaluate inequality and some Sobolev embeddings. One of the most difficult points is finding the appropriate functional spaces for the solution (ii) The second result is to investigating the regularized solution for our problem. We show the ill-posedness of the problem and give Fourier regularization. The most difficult thing that we have to overcome is finding the appropriate space, to prove that the regularized solution converges with the exact solution It can be said that our article is one of the first results, giving a general and comprehensive picture, considering both the frequency and the inaccuracy of Kirchhoff's diffusion equation with fractional time and space derivative. Using complex and interoperable assessment techniques, we find the right keys and tools to achieve both of our goals. This paper is organized as follows. In Section 3, we present the existence of the backward Problem (1) with the simple case F = 0. In the appropriate terms of the terminal data f , we show that the mild solution of (1) in the case β < 1 converges with the mild solution of the same problem in the case β = 1 when β ⟶ 1 − . Finally, in Section 4, we consider a backward problem with an inhomogeneous source term. The first part of this section discusses the existence of a mild solution under the appropriate conditions of the source function F. Furthermore, we also give an example, which shows that the problem is not stable, and then look for the approximate solution. Using the Fourier truncation method, we involve the regularized solution. Convergence error between the regularized solution and the correct solution has also been established, with some suitable conditions of input value data.

Preliminaries
If for each t > 0, the limitation finitely exists, then it is called the conformable derivative of order α ∈ ð0, 1 of v. We can refer the reader to [6,8,14,28,29]. We introduce fractional powers of A as follows: The space DðA ν Þ is a Banach space in the following with the corresponding norm: The information for negative fractional power A −ν can be provided by [30]. For any θ > 0, we introduce the following Hölder continuous space of exponent θ corresponding to the following norm: v For any 0 < θ < 1, let us introduce the following space: corresponding to the norm kvk C θ ðð0,T;BÞ ≔ sup 0<t≤T t θ kvð:,tÞk B .

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Let us define the space as follows:

Backward Problem for Homogeneous Case
In this section, we consider the final value problem for the homogeneous equation with a space fractional derivative as follows: where 0 < M 0 ≤ BðξÞ ≤ M 1 and ξ ∈ ½0, T. The following theorem states the existence and uniqueness of the solution of Problem (10).
Theorem 4. Let f ∈ X β,α ðΩÞ. Then, Problem (10) has a unique mild solution v ∈ Cð½0, T ; Furthermore, this solution is not stable in the L 2 norm.
Proof. We express a mild solution of (10) by Fourier series as follows: It follows from Problem (10) and the equality hð−ΔÞ β vð:,tÞ, w j i = λ β j hvð:,tÞ, w j i that Note that this formula is correct; we get that Multiply both sides of equation (15) by the quantity exp ð Ð tα/α 0 λ β j Bðk∇vð:,sÞk L 2 ÞdsÞ, we reach the following assertion: where we have used the fact that Integrating the two sides of the latter equation 0 to t, we obtain the following confirmation: v :,t ð Þ, w j exp It yields that v :,t ð Þ, w j = exp − Therefore, we find that For v ∈ L ∞ ð0, T ; H 1 ðΩÞÞ, we consider the following function: We shall prove by induction if w 1 , w 2 ∈ L ∞ ð0, T ; H 1 ðΩÞÞ, then 3 Journal of Function Spaces For m = 1, using the inequality je a − e b j ≤ ja − bj max ðe a , e b Þ for any a, b ∈ ℝ, we have Assume that (22) holds for m = k. We show that (22) holds for m = k + 1. Indeed, we have By the theory of the induction principle, (22) holds for all there exists a positive integer number k 0 such that Q k 0 is a contraction. It follows that the equation . Assume that f ∈ Xβ +γ,α ðΩÞ for any γ > β. Let us choose ν 0 such that where Proof. Let v α,β be the solution of Problem (11). Let w α be the solution to Problem (11) with β = 1.
Journal of Function Spaces whereby The term D 1 is bounded by The term D 2 is estimated as follows: Consider the following subset: If j ∈ A 1 , then using the inequality 1 − e −z ≤ C ε z ε , we get which allows us to obtain If j ∈ A 2 , then using the inequality Hence, we obtain Combining (36) and (38), we find that Let us choose 0 < ε < γ − β. Then, we follow from (33) and the latter equality that 5 Journal of Function Spaces This above inequality together with (32) and (3) yields that It is easy to get that It follows from the inequality we get The inequality ða + bÞ α ≤ a α + b α leads to which allows us to get immediately that It follows from (41) and (42) that for any t ∈ ½0, T Since the right-hand side of (47) is independent of t, we deduce that Then, we find that

Backward Problem for Inhomogeneous Case
In this section, we consider the final value problem for homogeneous equation as follows: Journal of Function Spaces where F is defined later.

Existence and Uniqueness of the Mild Solution.
In this subsection, we state the existence and uniqueness of the mild solution. In order to give the main results, we require the condition F which belongs to the space L 2p ð0, T ; Xβ, αðΩÞÞ.
Theorem 6. Let 0 < β ≤ 1 and F be the source function that belongs to L 2p ð0, T ; Xβ, αðΩÞÞ for any 1 < p < 1/ð1 − αÞ. Let B be the functions which satisfy M 0 ≤ BðzÞ ≤ M 1 , z ∈ ½0, T and Then, Problem (50) has a unique mild solution u ∈ L ∞ μ 0 ð0 , T ; H 1 ðΩÞÞ, where μ 0 is small enough. The function u satisfies that Furthermore, this solution is not stable in the L 2 norm.
Proof. By a simple calculation, we get the following equality: v By letting t = T and noting that vðx, TÞ = 0, we find that Let us denote by L ∞ μ ð0, T ; VÞ the functional subspace of L ∞ ð0, T ; VÞ corresponding to the norm where Set the following function: and we let So, using Parseval's equality, we get that

Journal of Function Spaces
Using the inequality je a − e b j ≤ ja − bj max ðe a , e b Þ, we continue to treat the term Mðs, t, j, w 1 Þ − Mðs, t, j, w 2 Þ as follows: Therefore, applying the Hölder inequality, we get Inserting (61) and (63) yields the following inequality: Take any δ ∈ ð0, 1Þ. By a similar explanation as (46), we find that By applying the Hölder inequality, we also obtain that From some observations as above, we deduce that Since the right-hand side of the latter estimate is independent of t, we find that Let us choose μ 0 such that Then, we can conclude that P is a contraction mapping in the space L ∞ μ 0 ð0, T ; H 1 ðΩÞÞ. Next, we continue to show Hence, from Parseval's equality, we find that This says that P v 1 belongs to the space L ∞ μ 0 ð0, T ; H 1 ðΩÞÞ. Using (68), we arrive at the confirmation that P v belongs to L ∞ μ 0 ð0, T ; H 1 ðΩÞÞ if v ∈ L ∞ μ 0 ð0, T ; H 1 ðΩÞÞ. For any m ∈ ℕ, let u m be the function that satisfies the following integral equation: 8 Journal of Function Spaces Let us assume that It is not difficult to verify that F m ∈ L ∞ ð0, T ; X β,α ðΩÞÞ, so we get that F m ∈ L 2p ð0, T ; X β,α ðΩÞÞ.
Using Theorem 6, we conclude that equation (72) has a unique solution u m ∈ L ∞ ð0, T ; H 1 ðΩÞÞ. By the fact that BðzÞ ≥ M 0 ∀z ∈ ℝ, we obtain the following estimate: The estimate is true for all t ∈ ½0, T, so it is easy to see that When m tends to +∞, we can check that k f m k L 2 ðΩÞ = 1/λ m go to zero when m ⟶ +∞ and This shows that Problem (50) is ill-posed in the sense of Hadamard in the L 2 -norm. ☐

Fourier Truncation Method.
In this section, we will provide a regularized solution and solve the problem by the Fourier truncation method as follows: Here, N ≔ NðδÞ goes to infinity as δ tends to zero which is called a parameter regularization. The function F is disturbed by the observed data F δ ∈ L ∞ ð0, T ; L 2 ðΩÞÞ provided by The main results of this subsection are given by the theorem below.

Theorem 7.
Let ν > 0 such that F belongs to the space L ∞ ð0 , T ; X β+ν,α ðΩÞÞ. Let F δ be as above. Let us assume that Problem (50) has a unique mild solution u ∈ L ∞ ð0, T ; DðA ν+θ ÞÞ for θ > 0. Let us choose N such that Here ν ≥ 1/2. Then, there exists a positive μ large enough such that Problem has a unique solution v N,δ ∈ L ∞ μ ð0, T ; DðA ν ÞÞ. Moreover, we have the following estimate: Remark 8. Since λ N~N 2/d , we can choose a natural number N such that Proof. Part 1: prove that the nonlinear integral equation (77) has a unique mild solution. Let any v ∈ Cð½0, T ; H 1 ðΩÞÞ, we denote by the following function

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By applying Parseval's equality, we follow from (82) that If 1 ≤ j ≤ N, then we have in view of (63) that The above two observations (65) lead to Because the right-hand side of the latter estimate is independent of t and noting the Sobolev embedding DðA ν Þ↪ H 1 ðΩÞ, we arrive at Let us choose μ 1 such that It is easy to see that G is a contracting mapping on the space L ∞ μ 1 ð0, T ; DðA ν ÞÞ. Therefore, we can conclude that there exists a uniqueness solution v N,δ for Problem (77).
Next, we continue to give the upper bound of the term kv N,δ ð·, tÞ − uð·, tÞk The above equality and Parseval's equality allow us to get that Since the condition BðzÞ ≤ M 1 ∀z ∈ ℝ and applying Hölder inequality, the quantity J 1 is bounded by where we have used the fact that kF δ − Fk L ∞ ð0,T;L 2 ðΩÞÞ ≤ δ.
The quantity J 2 is estimated as follows: This leads to the following estimate: The term J 3 is estimated as follows: Combining (89), (90), (93), and (94), we find that We choose μ such that both the following inequalities are satisfied: Some observations above give us the following confirmation: Since the fact that We easily obtain the desired result (80). ☐

Data Availability
No data were used to support this study.