Advances in Mathematical Physics
Volume 2021 (2021), Article ID 6370636, 8 pages
https://doi.org/10.1155/2021/6370636

Research Article

Existence for Time-Fractional Semilinear Diffusion Equation on the Sphere

N. D. Phuong,¹ Ho Duy Binh,² Ho Thi Kim Van,² and Le Dinh Long²

¹Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City 700000, Vietnam
²Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

Correspondence should be addressed to Le Dinh Long; ledinhlong@tdmu.edu.vn

Received 1 May 2021; Accepted 28 May 2021; Published 24 June 2021

Academic Editor: Mustafa Inc

Copyright © 2021 N. D. Phuong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Fractional diffusion on the sphere plays a large role in the study of physical phenomena customs and meteorology and geophysics. In this paper, we examine two types of the sphere problem: the initial value problem and the end value problem. We are interested in focus on the solution existence in a local or global form. In order to overcome difficult evaluations when evaluating, we need some new techniques. The main analytical tool is the use of the Banach fixed point theorem.

1. Introduction

When examining many physical and geophysical phenomena, one encounters problems directly or indirectly related to the sphere. To describe and explain quantitatively these models, they will be simulated with mathematical equations on the sphere. That is also the reason why the spherical equations have attracted many scientists interested and studied them. We can give some examples as follows. The weather forecasting models and the currents of groundwater in the ocean bed were simulated by equations on the sphere. For the readers’ convenience, we have given a number of typical works that have had a great influence on the development of the analysis of PDEs on the sphere.

The qualitative and numerical methods have been considered by many authors, such as Thong et al. [1, 2]. Cauchy problems for elliptical queries on spheres have been studied in [3, 4]. The Navier-Stokes equation on the 2D unit sphere has been considered by the recent paper [5]. Recently, pseudo-parabolic equation on the sphere has been studied in [6]. Intuitively, we realize that the root structure of the differential equation on the sphere can be complex, so studying the types of partial derivative equations on the sphere requires mathematical tools with many new techniques.

According to the development of the mathematics disciplines, especially calculus, in the last few decades, fractional analysis has been one of the most influential disciplines in mathematics. Most of the problems related to it often have applications in modeling real-world problems. Fraction analysis has many applications in mechanics, physics, engineering science, etc. We would like to share many published works on these issues such as Karapinar et al. [7–14] and Inc and his group [15–19].

The main goal of this paper is to investigate the existence of the following equation

with the initial Cauchy condition

or the terminal value condition

where called the Riemann-Liouville fractional derivative of order , . Here, is a sphere on the . It is given by is the Riemann-Liouville fractional derivative of order given by

and if . The functions in (1) are defined later. The operator is called Laplace-Beltrami which is introduced in more detail in Section 2. To the best of our knowledge, there are not any results on problem (1). Our main goal in this paper is to study two goals. Our first goal is to consider Cauchy initial problem (1)–(2). We get the global solution in a suitable space. Our second goal is to consider the Cauchy terminal problem (1)–(2). In this case, we get only local solutions in space . For both of the above purposes, we use Banach fixed point theorem together with the evaluation of the sphere. In addition, to overcome the difficulties of proofing, we also cleverly make use of subtle evaluations of the Mittag-Leffler functions. The two main difficulties shown are as follows:

(i)First, we deal with spherical harmonics on the sphere and require complex calculations(ii)Second, we must make sharp judgments for the Mittag-Leffler functions to achieve our goal

This paper is structured as follows. In Section 2, we introduce some preliminaries on Mittag-Leffler functions and their properties, Sobolev space on the sphere, and some other spaces. In Section 3, we focus on the initial value problem for problem (1)–(2). Section 4 provides the local well-posed result for terminal value problem (1)–(3).

2. Preliminaries

Mittag-Leffler is a function represented as the following form

, for and . We call to mind the following lemmas (see for example [20]). We have the following lemma which is useful for next proof.

Lemma 1. Let . Then, the function satisfies the following property

where and are the two positive constants.
Spherical harmonics are polynomials which satisfy (where is the Laplacian operator in ) and are restricted to the surface of the Euclidean sphere . The eigenvalues for in are

and the eigenfunctions corresponding to are the spherical harmonics of order , i.e.,

The space of all spherical harmonics of degree on , denoted by , has an orthonormal basis where

Let any function , so it is expressed by the expansion of spherical harmonics

where is the surface measure of the unit sphere. The Sobolev space is defined by

with the following norm

The space is defined by

with corresponding norm as follows

3. Global Existence for Mild Solution to Initial Value Problem

Theorem 2. Assume that such that

Let satisfies the condition

where is a positive constant and and satisfy that

Then, problem (1)–(2) has a global existence in for enough large.

Proof. As we know from [2] that is the Laplace-Beltrami on the sphere . Any function can be described by the terms of spherical harmonics

where is the surface measure of the unit sphere. Let us first give an expression of the mild solution.

So, we get that the following equality

Set the following function

Setting the function

Noting that for any , and , the first term is bounded by

which allows us to get that

for any . This gives immediately that if then belongs to the space . Let us take two functions and belong to the space . Then, from (21), we have

Using Parseval’s equality and Hölder inequality, one has

Let us review that , we note that the following inequality

for . Noting that we get that the following inequality

Since the assumption and , we can deduce that

Here, is beta function. Let us choose ; we see that satisfies . Combining (26) and (28), we provide that

where we set

Thanks to Lipschitz property of , we find that for any

Due to the condition we know that the following estimate

This implies immediately that

The right hand side of (34) is independent of , so we can deduce that the following estimate

By choose enough large such that , we find that is a contraction in . Based on the Banach fixed point theorem, we have immediately concluded that problem (1)–(2) has a global existence in . ☐

4. Terminal Value Problem: Local Existence

In this section, we devoted the following problem with terminal condition

where is defined later. The purpose of this section is to study the existence and uniqueness of solution of problem (1)–(3).

Theorem 3. Let us assume that such that

Let the function . Let satisfies the condition

for any . Then, problem (1)–(3) has a local existence in .

Proof. Let us assume that . Then, we have

Set into the above equation, we get that

This implies that

After some simple calculation, we have that

So, we get that the following equality

In order to apply Banach fixed point theorem, we need to set the following function

First, let us look at the expression as above and give an evaluation for for any . First, noting that

we get that

This implies immediately that

where we have used Hölder inequality. Using (46), we have that

Hence, we can deduce that

Lipschitz property of as in (38) gives that

Combining (49) and (50), we find that

which allows us to get that

The term to the right of above expression is independent of , so we get that

By a similar argument as above, we deduce that

where in the last inequality, we have used Hölder inequality. Let us repeat that(14)and , and thanks to that, we immediately have the following two estimations

Combining (54) and (55), we arrive at

The right hand side of (56) is independent of ; we have the following conclusion immediately

From these two assertions (44), (53), and (57) and using the triangle inequality, we have the following affirmation

By choose enough small, we can conclude that is a contraction. Next, we need to check that . Indeed, we get for any

Using Banach fixed point theorem, we can conclude that has a fixed point in . Hence, we get the desired result. ☐

5. Conclusion

In this paper, this is one of our first results about fractional diffusion on the sphere. In this article, we are interested in the existing existence in local form and global format with the main tool is the Banach fixed point theorem. In the future, we study the ill-posedness of this problem and show the convergent rate between the sought solution and the regularized solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

Both authors contributed equally and significantly in writing this paper. Four authors read and approved the final manuscript.

Acknowledgments

This research is supported by the Industrial University of Ho Chi Minh City (IUH) under grant number 66/HD-DHCN.

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