# Existence for Time-Fractional Semilinear Diffusion Equation on the Sphere

Authors: N. D. Phuong; Ho Duy Binh; Ho Thi Kim Van; Le Dinh Long
Journal: Advances in Mathematical Physics
 (2021)

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## 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.


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## Body

## 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(1)∂∂tux,t−∂1−α∂tΔ∗ux,t=θtFux,t,x,t∈S2×0,T,with the initial Cauchy condition
(2)ux,0=fx,or the terminal value condition
(3)ux,T=gx,where∂α/∂tα called the Riemann-Liouville fractional derivative of order α, 0<α≤1. Here, S2 is a sphere on the R3. It is given by ∂1−α/∂t is the Riemann-Liouville fractional derivative of order 1−α given by
(4)∂1−α∂tvt≔1Γαddt∫0tsα−1vt−sds,0<α<1,andD0+0.02cmαvt≕d/dtvt if α=1. The functions ψ,F in (1) are defined later. The operator Δ∗ 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 L∞. 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 goalThis paper is structured as follows. In Section2, 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(5)Eα,βξ=∑n=0∞ξnΓnα+β,ξ∈ℂ, for α>0 and β∈ℝ. We call to mind the following lemmas (see for example [20]). We have the following lemma which is useful for next proof.Lemma 1.
Let0<α<1. Then, the function Eα,1−z satisfies the following property
(6)C11+z≤Eα,1−z≤C21+z,z>0,
whereC1 and C2 are the two positive constants.
Spherical harmonics are polynomials which satisfyΔxYx=0 (where Δx is the Laplacian operator in R3) and are restricted to the surface of the Euclidean sphere S2. The eigenvalues for −Δ∗ in R3 are
(7)λj=j2+j,j=0,1,2,⋯⋯,
and the eigenfunctions corresponding toλj are the spherical harmonics Wjx of order l, i.e.,
(8)Δ∗Wjx=−λjWjx.
The space of all spherical harmonics of degreej on S2, denoted by Vj, has an orthonormal basis Wjkx:k=1,2,3,⋯N2,j where
(9)N2,0=1,N2,j=2j+1Γ2,j≥1.
Let any functionf∈L2S2, so it is expressed by the expansion of spherical harmonics
(10)f=∑j=0∞∑k=12j+1/Γ2f^jkWjk,f^jk=∫S2fW¯jkdS,
wheredS is the surface measure of the unit sphere. The Sobolev space HνS2 is defined by
(11)HνS2=g∈L2S2:∑j=0∞∑k=12j+1/Γ2j2+j+1νg∧jk2<∞,with the following norm
(12)gHνS2=∑j=0∞∑j=12j+1/Γ2j2+j+1νg∧jk2.
The spaceLa∞0,T;HνS2 is defined by
(13)La∞0,T;HνS2=f∈L∞0,T;HνS2,e−atf.,tHνS2<∞,with corresponding norm as follows
(14)fLa∞0,T;HνS2=sup0≤t≤Te−atf.,tHνS2.


## 3. Global Existence for Mild Solution to Initial Value Problem

Theorem 2.
Assume thatθ:0,T⟶ℝ such that
(15)∣θz∣≤Cθzm,−12<m.
LetF satisfies the condition
(16)Fv1−Fv2Hν′S2≤Kfv1−v2HνS2,
whereKf is a positive constant and ν and ν′ satisfy that
(17)0<ν−ν′<min2,1α.
Then, problem (1)–(2) has a global existence in La∞0,T;HνS2 for a enough large.Proof.
As we know from [2] that Δ∗ is the Laplace-Beltrami on the sphere S2. Any function u∈L2S2 can be described by the terms of spherical harmonics
(18)ux,t=∑j=0∞∑k=12j+1/Γ2u^jktWjkx,u^jkt=∫S2ux,tW¯jkxdS,
wheredS is the surface measure of the unit sphere. Let us first give an expression of the mild solution.
(19)u^jkt=Eα,1−j2+jtαf^jk+∫0tEα,1−j2+jt−rαθrF^jkurdr.
So, we get that the following equality(20)ux,t=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαf^jkWjkx+∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF^jkurdrWjkx.
Set the following function(21)Jwt=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαf^jkWjkx+∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF^jkwrdrWjk.
Setting the function(22)M1x,t=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαf^jkWjkx.
Noting thatEα,1−z≤C2 for any z>0, and f∈HνS2, the first term is bounded by
(23)∥M1.,t∥HνS22=∑j=0∞∑k=12j+1/Γ2j2+j+1νEα,1−j2+jtαf∧jk2≤C22∑l=0∞∑k=12j+1/Γ2j2+j+1νf∧lk2=C22∥f∥HνS22,which allows us to get that
(24)∥M1.,t∥HνS2≤C2∥f∥HνS2,
for any0≤t≤T. This gives immediately that if w=0 then Jw belongs to the space L∞0,T;HνS2. Let us take two functions w and w¯ belong to the space L∞0,T;HνS2. Then, from (21), we have
(25)Jwt−Jw¯t=∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF^jkwr−F^jkw¯rdrWjkx.
Using Parseval’s equality and Hölder inequality, one has(26)∥Jwt−Jw¯t∥HνS22=∑j=0∞∑k=12j+1/Γ2j2+j+1ν∫0tEα,1−j2+jt−rαθrF∧jkwr−F∧jkw¯rdr2≤∑j=0∞∑k=12j+1/Γ2j2+j+1ν∫0tEα,1−j2+jt−rαθr2dr∫0tF∧jkwr−F∧jkw¯r2dr.
Let us review thatEα,1−z≤C2/1+z, we note that the following inequality
(27)Eα,1−j2+jtα≤C21+j2+jtα≤C2j2+j−βt−αβ,
for0<β<1. Noting that j2+j≥j2+j+1/2 we get that the following inequality
(28)j2+j+1ν∫0tEα,1−j2+jt−rαθr2dr≤C22Cθ24−βj2+j+1ν−2β∫0tt−r−2αβr2mdr.
Since the assumption1−2αβ>0 and 1+2m>0, we can deduce that
(29)∫0tt−r−2αβr2mdr=t1−2αβ+2mB1−2αβ,1+2m.
Here,B is beta function. Let us choose β=1/2ν−ν′; we see that β satisfies 1−2αβ>0. Combining (26) and (28), we provide that
(30)∥Jwt−Jw¯t∥HνS22≤C~12∑j=0∞∑k=12j+1/Γ2j2+j+1ν−2β∫0tF∧jkwr−F∧jkw¯r2=C~12∫0t∥Fwr−Fw¯r∥Hν−2βS22dr=C~12∫0t∥Fwr−Fw¯r∥Hν′S22dr,where we set
(31)C~12=C22Cθ24−βB1−2αβ,1+2m.
Thanks to Lipschitz property ofJ, we find that for any a>0(32)e−2at∥Jwt−Jw¯t∥HνS22≤C~12Kf2T1−2αβ+2m∫0te−2at−re−2ar∥wr−w¯r∥HνS22dr≤C~12Kf2T1−2αβ+2m∫0te−2at−rdr∥w−w¯∥La∞0,T;HνS22.
Due to the condition∫0te−2at−rdr≤1/2a, we know that the following estimate
(33)e−2at∥Jwt−Jw¯t∥HνS22≤C~12Kf2T1−2αβ+2m2a∥w−w¯∥La∞0,T;HνS22.
This implies immediately that(34)e−at∥Jwt−Jw¯t∥HνS2≤∣~C12Kf2T1−2αβ+2m2a∥w−w¯∥La∞0,T;HνS2.
The right hand side of (34) is independent of t, so we can deduce that the following estimate
(35)∥Jw−Jw¯∥La∞0,T;HνS2≤C~12Kf2T1−2αβ+2m2a∥w−w¯∥La∞0,T;HνS2·
By choosea enough large such that C~12Kf2T1−2αβ+2m/2a<1, we find that J is a contraction in La∞0,T;HνS2. Based on the Banach fixed point theorem, we have immediately concluded that problem (1)–(2) has a global existence in La∞0,T;HνS2. ☐


## 4. Terminal Value Problem: Local Existence

In this section, we devoted the following problem with terminal condition(36)∂∂twx,t−∂1−α∂tΔ∗wx,t=θtFux,t,x,t∈S2×0,T,wx,T=gx,x∈S2,whereg 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 thatψ:0,T⟶ℝ such that
(37)∣θz∣≤Cθzδ,δ>−1.
Let the functiong∈H2S2. Let F satisfies the condition
(38)∥Fv1−Fv2∥L2S2≤Kf∥v1−v2∥L2S2,
for anyKf>0. Then, problem (1)–(3) has a local existence in L∞0,T;L2S2.Proof.
Let us assume thatux,0=fx. Then, we have
(39)w^jkt=Eα,1−j2+jtαf^jk+∫0tEα,1−j2+jt−rαθrF^jkwrdr.
Sett=T into the above equation, we get that
(40)w^jkT=Eα,1−j2+jTαf^jk+∫0TEα,1−j2+jT−rαθrF^jkwrdr.
This implies that(41)f^jk=1Eα,1−j2+jTαg^jk−∫0TEα,1−j2+jT−rαθrF^jkwrdr.
After some simple calculation, we have that(42)w^jkt=Eα,1−j2+jtαEα,1−j2+jTαf^jk+∫0tEα,1−j2+jt−rαθrF^jkwrdr−Eα,1−j2+jtαEα,1−j2+jTα∫0TEα,1−j2+jT−rαθrF^jkwrdr.
So, we get that the following equality(43)wx,t=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTαg^jkWjkx+∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF^jkwrdrWjkx−∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTα∫0TEα,1−j2+jT−rαθrF^jkwrdrWjkx.
In order to apply Banach fixed point theorem, we need to set the following function(44)Tθt=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTαg^jkWjk⏟T0t+∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF^jkθrdrWjk⏟T1θt−∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTα∫0TEα,1−j2+jT−rαθrF^jkθrdrWjk⏟T2θt.
First, let us look at the expression as above and give an evaluation forT2θ1t−T2θ2t for any θ1,θ2∈L∞0,T;L2S2. First, noting that
(45)Eα,1−j2+jTα≥C11+j2+jTα,Eα,1−j2+jT−rα≤C21+j2+jT−rα,we get that
(46)Eα,1−j2+jT−rαEα,1−j2+jTα≤C2C11+j2+jTα1+j2+jT−rα≤C3TαT−r−α.
This implies immediately that(47)∥T2θ1t−T2θ2t∥L2S22=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTα∫0TEα,1−j2+jT−rαθrF∧jkθ1r−F∧jkθ2rdr2≤∑j=0∞∑k=12j+1/Γ2∫0TEα,1−j2+jT−rαEα,1−j2+jtαEα,1−j2+jTαθrdr∫0TEα,1−j2+jT−rαEα,1−j2+jtαEα,1−j2+jTαθrF∧jkθ1r−F∧jkθ2r2dr,where we have used Hölder inequality. Using (46), we have that
(48)∫0TEα,1−j2+jT−rαEα,1−j2+jtαEα,1−j2+jTαθrdr≤C3C1TαCθ∫0TT−r−αrδdr=C3C1TαCθT1−α+δB1−α,1+δ=C3C1CθT1+δB1−α,1+δ,∫0TEα,1−j2+jT−rαEα,1−j2+jtαEα,1−j2+jTαθrF∧jkθ1r−F∧jkθ2r2dr≤C3C1TαCθ∫0TT−r−αrδF∧jkθ1r−F∧jkθ2r2dr.
Hence, we can deduce that(49)∥T2θ1t−T2θ2t∥L2S22≤C3C1Cθ2T1+δ+αB1−α,1+δ∫0TT−r−αrδ∥Fθ1r−Fθ2r∥L2S22dr.
Lipschitz property ofF as in (38) gives that
(50)∫0TT−r−αrδ∥Fθ1r−Fθ2r∥L2S22dr≤Kf∫0TT−r−αrδ∥θ1.,r−θ2.,r∥L2S22dr≤Kf∫0TT−r−αrδdr∥θ1−θ2∥L∞0,T;L2S22=KfT1−α+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S22.
Combining (49) and (50), we find that
(51)∥T2θ1t−T2θ2t∥L2S22≤C3C1Cθ2T2+2δB1−α,1+δ2∥θ1−θ2∥L∞0,T;L2S22,which allows us to get that
(52)∥T2θ1t−T2θ2t∥L2S2≤C3C1CθT1+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S2.
The term to the right of above expression is independent oft, so we get that
(53)∥T2θ1−T2θ2∥L∞0,T;L2S2≤C3C1CθT1+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S2.
By a similar argument as above, we deduce that(54)∥T1θ1t−T1θ2t∥L2S22=∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrF∧jkθ1r−F∧jkθ2rdr2≤∑j=0∞∑k=12j+1/Γ2∫0tEα,1−j2+jt−rαθrdr∫0tEα,1−j2+jt−rαθrF∧jkθ1r−F∧jkθ2r2dr,where in the last inequality, we have used Hölder inequality. Let us repeat thatEα,1−j2+jt−rα≤C2,(14)and θr≤Cθrδ, and thanks to that, we immediately have the following two estimations
(55)∫0tEα,1−j2+jt−rαθrdr≤C2Cθ∫0trδdr=C2Cθt1+δ1+δ≤C2CθT1+δ1+δ·
Combining (54) and (55), we arrive at
(56)∥T1θ1t−T1θ2t∥L2S22≤C2CθT1+δ1+δC2Cθ∫0trδ∥Fθ1r−Fθ2r∥L2S22dr≤Kf2C2CθT1+δ1+δC2Cθ∫0trδ∥θ1.,r−θ2.,r∥L2S22dr≤Kf2C2CθT1+δ1+δC2Cθ∫0trδdr∥θ1−θ2∥L∞0,T;L2S22≤KfC2CθT1+δ1+δ2B1−α,1+δ∥θ1−θ2∥L∞0,T;L2S22.
The right hand side of (56) is independent of t; we have the following conclusion immediately
(57)∥T1θ1−T1θ2∥L∞0,T;L2S2≤KfC2CθT1+δ1+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S2.
From these two assertions (44), (53), and (57) and using the triangle inequality, we have the following affirmation
(58)∥Tθ1−Tθ2∥L∞0,T;L2S2≤∥T1θ1−T1θ2∥L∞0,T;L2S2+∥T2θ1−T2θ2∥L∞0,T;L2S2≤C3C1CθT1+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S2+KfC2CθT1+δ1+δB1−α,1+δ∥θ1−θ2∥L∞0,T;L2S2.
By chooseT enough small, we can conclude that T is a contraction. Next, we need to check that T0tg∈L∞0,T;L2S2. Indeed, we get for any 0≤t≤T(59)∥T0tg∥L2S22=∑j=0∞∑k=12j+1/Γ2Eα,1−j2+jtαEα,1−j2+jTα2g∧jk2≤∑j=0∞∑k=12j+1/Γ21+j2+jTαC12f∧jk2≤C~∥g∥H2S22.
Using Banach fixed point theorem, we can conclude thatT has a fixed point w in L∞0,T;L2S2. 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.



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