On the Convergence Result of the Fractional Pseudoparabolic Equation

In this paper


Introduction
In this paper, we consider a pseudoparabolic equation with fractional Laplacian defned as follows: where Ω denotes the domain of the spatial variable x, zΩ is the boundary of Ω, and k > 0 is a constant coefcient.Te function G is the nonlinear source term which appears in some physical phenomena and v(x, t) describes the state of the unknown function generally at position x and time t.
Our main goal in this paper is to study the convergence of the mild solution of problem 1.1 with the initial condition as follows: In this paper, for simplicity, we only consider Ω � (0, π) and so zΩ can be understood by the collection of 2 discrete points zΩ � 0, π { }.Te notation (−∆) s is called fractional Laplacian with order 0 < s ≠ 1.From the abovementioned reasons, we can call this equation the "1-dimensional nonlinear fractional Laplacian pseudoparabolic equation." As for k � 0, we obtain the classical nonlinear parabolic equation.Moreover, when s � 0, we obtain the ordinary pseudoparabolic equation.Both of these equations had been carefully studied by many researchers recently [1][2][3].
Pseudoparabolic equations have been studied extensively in recent years.It describes a variety of physical phenomena and also has applications in many diferent felds.One of the debates that have taken place in relation to equation ( 1) is about the local fractional operator (−∆) s , in which many researchers believe that many physical phenomena are better described compared to the classical integral diferential equation.For more information on this regard and the properties of the operator (−∆) s , refer to references [4][5][6][7][8][9].
Te study of fractional pseudoparabolic equations has always attracted the attention of many researchers because of their various applications in diferent felds, such as unidirectional propagation of long waves in a nonlinear dispersed medium, homogeneous liquid permeability in fractured rock, and heat conduction involving two temperatures [10,11].Let us mention some previous results on fractional pseudoparabolic equations.In [4], the researchers carried out on the fractional parabolic equation by considering the Cauchy problem of this equation in the whole space R n .In this work, the authors have investigated the global existence and time-decay rates for small-amplitude solutions.Some researchers have also studied the semilinear pseudoparabolic equation with Caputo derivative [1,12], a fnal boundary value problem for a class of fractional pseudoparabolic with a nonlinear reaction term [3], and a nonlinear Kirchhof's model of the pseudoparabolic type in references [13,14].In these works, frst, the existence, uniqueness, and regularity of the mild local solution have been investigated.Next, the stability and regularity of the solution are studied and discussed.Te main techniques and methods frequently used are the modifed Lavrentiev regularization method and the Fourier truncated regularization method.To the fractional pseudoparabolic equation, sometimes, the inverse source problem is also discussed [15][16][17].
In [13], the authors considered a nonlinear Kirchhof's model of pseudoparabolic type.Tey obtained the results on the local existence and regularity of mild solution.Te authors also showed that the ill-posed property in the sense of Hadamard of the problem when the fractional order is larger than 1.By using the Fourier truncation method to regularize the problem, they established some stability estimates on the H p norm under some a-priori conditions on the sought solution.
Recently, in [15], the authors focused on the source problem for the pseudoparabolic equation with fractional Laplacian.In this article, they also investigated the convergence of the source function when the fractional order tends to 1 − .Tere are not many results devoted to the convergence of mild solution v(x, t) when the fractional order tends to 1 − .Motivated by the results in [15], we decided to study the fractional pseudoparabolic equation (1) and investigate the convergence of the state function v(x, t) when the fractional order tends to 1 − .
In the following section, we present a brief overview of this work.Te next section gives some preliminary knowledge on used notations, the spectral analysis of Laplacian the defnition of fractional Laplacian, and some information about the functional space of interest.Section 3 is dedicated to the calculation of the explicit formula of mild solution to problem 1.1.Section 4 is to investigate the convergence of the mild solution concerning its fractional order in the Laplacian operator when s tends to 1 − .Tis result is important because of the relationship between the physical phenomena involved in equations when s � 1 and s < 1.By letting s tend to 1 − , we observe the association of subdifusion phenomena with normal difusion.Te last section gives some discussion and proposes some directions for improving.

Te Less Tan or Equivalent To Notation.
Given two positive quantities y and z, we write y ≲ z if there exists a constant C > 0 such that y ≤ Cz.

Relevant Notation.
Let us recall the following spectral problems: which admit a family of eigenvalues 0 We also notice that the collection of eigenfunction e j (x) could form an orthonormal basis of L 2 (Ω).In this paper, the domain of the spatial variable is Ω � (0, π), and we can directly calculate the eigenvalues λ j � j 2 for j � 1, 2, 3, . . .along with the eigenfunctions e j (x) � ��� 2/π √ sin(jx).But for more convenience, we sometimes reuse these symbols e j (x) and λ j in next steps.

Te Mittag-Lefer Function
Defnition 1 where α > 0 and β ∈ R are arbitrary constants and Γ is the Gamma function.

Te Fractional Laplacian Operator and Inner Product.
For s ≥ 0, we defne by (−∆) s the following operator: 〈v, e j 〉λ s j e j , (5) and the inner product is defned as follows: We recall the Hilbert scale space, which is given as follows: (7) for any s ≥ 0. It is well known that H s (Ω) is a Hilbert space corresponding to the norm as follows: 2 Journal of Mathematics , f ∈ H s (Ω). (8)
Te frst equation of ( 10) is a diferential equation with the classical derivative as follows: It is easy to see that the solution of it as follows: For simplicity, we denote f j � 〈f, e j 〉 and G j � 〈G(., t), e j 〉 and obtain the formula as follows: Lemma 2. Te mild solution to NFLPPE ( 1) and ( 2) is given by the following formula: Journal of Mathematics where Lemma 3. Te mild solution to NFLPPE ( 1) and ( 2), when s � 1 is given by the following formula: Lemma 4 (Weakly singular Gronwall's inequality, see [18]).

Main Results
Te existence result of the mild solution to problem 1.1 is widely and carefully discussed in Teorem 4.1 of [19] when the nonlinear term G is the global Lipschitz and satisfed some particular conditions, so we ignore that part and focus only in investigating the convergence of the mild solution while s ⟶ 1 − .Te obtained result is fully presented by the following theorem.

4
Journal of Mathematics (M 1 ): from [15], we get that if j ≥ 1, then we fnd that j 2s − j 2 ≤ C θ j s+θ (1 − s) θ , for any θ > 0 and C θ is the constant which depends on θ.For any μ > 0, in view of the inequality |e Let us choose s, θ such that s + θ ≤ 2. Ten, we obtain exp − Terefore, and this means that M 1 is bounded.(M 2 ): Using the Hölder inequality, we fnd that Due to the e

Tis implies that
Journal of Mathematics Since ρ ≤ 2 sε and ε ≤ 1, we obtain that Since the global Lipschitz is G, we derive that where we have used the Sobolev embedding H ρ (Ω)↪L 2 (Ω).(M 3 ): applying the Hölder inequality, we infer that and using (22), we obtain that exp − j 2s (t − r) Tus, we obtain (33) From (30), we have where C(μ, θ, K g , k) indicates the constant which depends on μ, θ, K g , k.It is easy to see that  t 0 (t − r) 2μ dr � t 1+2μ /1 + 2μ.Hence, we infer that ( Combining three steps as mentioned earlier, we derive that

Conclusion
Motivated by the result of [15], this work has considered the 1-dimensional fractional Laplacian pseudoparabolic equation with the nonlinear source term.By considering the problem included with an initial condition and some conditions in the source function, we showed the continuous dependence of mild solution to the fractional-order parameter.