On a nonlinear volterra integrodifferential equation involving fractional derivative with Mittag-Leffler Kernel

ON A NONLINEAR VOLTERRA INTEGRODIFFERENTIAL EQUATION INVOLVING FRACTIONAL DERIVATIVE WITH MITTAG-LEFFLER KERNEL TOM´ AS CARABALLO, TRAN BAO NGOC, NGUYEN HUY TUAN, AND RENHAI WANG Abstract. In this paper, a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces is studied. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, the existence of a mild solution of our problem is proved. The main tool to prove our results is the use of some Sobolev embeddings. Keywords: Riemann-Liouville fractional derivative, Time diffusion equation, well–posedness, regularity estimates. 2010 MSC: 26A33, 35B65,35B05, 35R11. 1. Introduction During the last decades, fractional calculus has become a powerful tool with accurate and successful results in modeling several complex phenomena in many various fields of science and engineering. One of the branches of fractional calculus is the theory of fractional diffusion equations. Timefractional diffusion equations open up great opportunities to model challenging phenomena such as long-range time memory or spatial interactions, nonlocal and local dynamics. For more details, see [1, 3, 38, 40] and the references therein. One of the modern trends in fractional calculus is the development of fractional operators with non-singular kernels. Studying new fractional derivatives with different singular or nonsingular kernels is important in order to satisfy the need for applied modeling in various fields, such as fluid mechanics, viscoelasticity, biology, physics and engineering [16, 17]. Some definitions of fractional derivatives were given based on nonsingular kernels such as the Atangana–Baleanu fractional derivatives. Recently, many fractional models with non-singular kernel are receiving an increasing interest of many researchers with some different research directions. The special fact of Atangana–Baleanu derivative is that it possesses Mittag– Leffler function as its kernel, which is non-local as well as non-singular. The importance of fractional derivatives with non-singular kernels are particularly oriented to models of dissipative phenomena which cannot be adequately described by the classical fractional derivatives [11, 12, 13]. In [19], the authors considered a new chaotic model in two fractional operators, that is, the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative. In [22], the authors considered a comparison study of bank data with different fractional operators such as Caputo, Caputo–Fabrizio and Atangana–Baleanu. They also proved that the results of the fractional Atangana–Baleanu operator is more accurate and flexible. In [23], modeling the transmission of dengue infection is introduced by using Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) fractional derivatives. Until now, to the best of our knowledge, the works on analysis existence and regularity for ODEs and PDEs with Atangana–Baleanu derivative is very limited. In [18], the authors show the existence of the Keller-Segel model with Caputo and Atangana-Baleanu fractional derivative using fixed point theory. In [24], the fractional logistic model concerned with Atangana–Baleanu fractional derivative is considered. This paper studies time fractional Volterra integro-differential equation with nonlinear source as follows Dα tu(x, t) + Aβ/2u(x, t) = Zt 0 R(t, τ)F(u(x, τ))dτ, (x, t)∈(0, T)×Ω,(1) adressed with the Dirichlet boundary condition u(·, t)∂Ω= 0, t ∈(0, T),(2) and the initial value condition u(x, 0) = φ(x), x ∈Ω,(3) 1 where Dα tis the Atangana-Baleanu fractional derivative of order αin Caputo sense (see Definition (2.1)). Existence and regularity for fractional diffusion equations with Caputo derivative has also been studied by several authors. Regularity theory enables us to improve the smoothness and stability of solutions in various solution spaces and this leads to efficient ways for numerical simulations. For fractional diffusion equations with time fractional derivative in the sense of Caputo, Riemann-Liouville, etc, some authors developed and obtained interesting results. Carvalho et al. [29] established a local theory of mild solutions where Ais a sectorial (nonpositive) operator. Guswanto [30] studied the existence and uniqueness of a local mild solution to a class of initial value problems for nonlinear fractional evolution equations. Besides, The existence, uniqueness and regularity of solutions are established in some previous works see, for instance, [26, 27, 28]. As for semilinear Volterra integrodifferential equations with integer order derivative, we can list some interesting works. In 1988, Heard and Rakin [31], considered the following Volterra integro-differential equation ∂tu+A(t)u(t) = Rt 0R(t, τ)F(u(τ))dτ, t ∈(0, T), u(0) = u0.(4) The authors studied in [32] the fractional Volterra integro-differential equation with ψ-Hilfer fractional derivative. In [33] it is proved the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Rashid and Qaderi [34], established the local and global existence of mild solutions to a class of fractional integro-differential equations in an arbitrary Banach space. Rashid and Al-Omari [35] studied the local and global existence of mild solutions to a class of fractional semilinear impulsive Volterra type integro-differential evolution equations. In 2017, Gou and Li [36] studied local and global existence of mild solution for an impulsive fractional functional integro differential equation with non-compact semi-group in Banach spaces. Chen et al. [39] considered the following fractional non-autonomous integro-differential evolution equation of Volterra type in Banach space ∂α tu+A(t)u(t) = Rt 0R(t, τ)F(u(τ))dτ +G(t, u(t)), t ∈(0, T), u(0) = u0.(5) where ∂α tis the standard Caputo fractional time derivative of order 0 < α ≤1. They first proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, they obtained a blow-up alternative result. Our new results in this paper are described as follows •Our first goal is to establish the global existence for integro-differential evolution equation with fractional derivative. To overcome some difficulties, we introduce some weighted Lebesgue spaces. The key tool for our analysis here is the techniques on Kummer/hypergeometric function in Garrido-Atienza et al. [2, Lemma 8]. Hence, we can overcome some challenge estimates and obtain the global well-posed result. •Next. the study on the regularity property for PDEs result in LpSobolev spaces is interesting and still an open direction. Until now, there are very few papers on Lpregularity for fractional evolution equation. As we know, regularity estimates for Lpspaces are key ingredients to prove the existence and uniqueness of very weak solutions of some classes of elliptic equations. The second new result in the present paper is to investigate the weighted Lpestimate for the mild solution when the initial datum φbelongs to LqSobolev space. To our knowledge, there are no previous results of this type for fractional diffusion with Atangana-Baleanu fractional derivative. The proof of our results is based on the Sobolev embedding theorems and some fixed point theorems. The content of our paper is organized as follows. In Section 2, we recall some notation, definitions, and preliminary results regarding the solution representation and establish a definition of mild solution. In Section 3, we prove the well-posedness of a nonlinear time-fractional Volterra equation with the Atangana-Baleanu fractional derivative. In Section 4, we discuss an application of our main results to an initial value problem for a time-space Volterra equation. 2. Preliminaries We will recall in this section some notation, definitions, and preliminary results concerning the solution representation and establish an appropriate definition of mild solution. We will structure the section in several subsections for more clarity. 2 2.1. ODEs with the Atangana-Baleanu fractional derivative in Caputo’s sense. Let us first recall the following definition. Definition 2.1. For ϕ∈H1(0, T ), its Atangana-Baleanu fractional derivative in Caputo’s sense of order αis given by Dα tϕ(t) = ωα bαZt 0 ∂ϕ(τ) ∂τ Eα,1(−λα(t−τ)α)dτ, (6) where bα= 1 −α,ωα=bα+α(Γ(α))−1, and λα=bα−1α. Next we recall the following result. Lemma 2.2. Solutions of the initial value problem Dα tf(t) + κf(t) = G(f(t)), t ∈(0, T), f(0) = f0,(7) are given by f(t) = ηEα,1(−ρtα)f0+bα ωα η G(f(t)) + µZt 0 (t−τ)αEα,α(−ρ(t−τ)α)G(f(τ))dτ, where the numbers η,ρ, and µare formulated as η=ωα ωα+bακ ρ=ακ ωα+bακ, µ =ωα−bα ωα η+bα ωα ηρ. Proof. This type of fractional differential equation involving the fractional-time derivative with nonsingular Mittag-Leffler kernel has been solved in [9] by the means of Laplace transform.  For more details of Caputo fractional time-derivative and the fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, we refer to recent interesting papers [42, 43, 44]. 2.2. Kummer/hypergeometric function. Using the factor e−mt with large enough number m > 0 plays an important role in establishing the global well-posedness. This makes the appearance of the so-called Kummer function or hypergeometric function, which is defined by K(a, b, m) := Γ(b) Γ(b−a)Γ(a)Z1 0 (1 −τ)b−a−1τa−1emτ dτ, b > a > 0, m ∈C. A nice property of this function on its asymptotic behavior is the following: K(a, b, m) := Γ(b)(Γ(a))−1emm−(b−a)1 + O1 |m|, as can be seen in Garrido-Atienza et al. [2, Lemma 8]. For each t > 0, the above property can be scaled from the interval (0,1) to (0, t) by using a simple substitution. For m > 0, it may be checked that Zt 0 (t−τ)a−1τb−1emτ dτ =ta+b−1Zt 0 (1 −τ)a−1τb−1emtτ dτ =ta+b−1Γ(a)Γ(b) Γ(a+b)K(b, a +b, mt) =ta+b−1emtΓ(a) (mt)a1 + O1 mt =tb−2m−aemtΓ(a)t+O1 m.(8) More details of this function can be found in [8]. 2.3. Solution representation. In this paper, we consider the symmetric uniformly elliptic operator A:H2(Ω)∩H1 0(Ω) ⊂L2(Ω) →L2(Ω) defined by Aϕ(x) = −PN i=1 ∂ ∂xiPN j=1 Aij(x)∂ ∂xjϕ(x)+b(x)ϕ(x). Here, we suppose that Aij =Aji,1≤i, j ≤Nand there exists a constant Λ >0 such that for all (ξ1, ξ2, ..., ξN)∈RNand Λ PN i=1 ξ2 i≤P1≤i,j≤NAij(x)ξiξj,for x∈Ω∪∂Ω.Suppose, furthermore, that Aij ∈C1(Ω ∪∂Ω), b∈C(Ω ∪∂Ω; R+). Then, the spectrum and the corresponding eigenvectors of Aare given by 0 < λ1≤λ2≤... ≤λn% ∞ and b1, b2, ..., bn, ... ⊂H2(Ω) ∩H1 0(Ω). Note that {bn}forms an 3 orthonormal basis of L2(Ω). In order to find a formula for solutions, we firstly write equation of (1) in the n-th dimension as follows Dα tun(t) + λβ/2 nun(t) = Zt 0 R(t, τ)Fn(u(·, τ))dτ, t ∈(0, T), which is equipped with the initial value condition un(0) = φn. By applying Lemma 2.2, solutions of the above system of ODEs are given by un(t) = bα ωα ηnZt 0 R(t, τ)Fn(u(·, τ))dτ +ηnEα,1(−ρntα)φn +µnZt 0Zt0 0 (t−t0)αR(t0, τ)Eα,α(−ρn(t−t0)α)Fn(u(·, τ))dτdt0,(9) where the numbers ηn,ρn, and µnare formulated by ηn=ωα ωα+bαλβ/2 n ρn=αλβ/2 n ωα+bαλβ/2 n , µn=ωα−bα ωα ηn+bα ωα ηnρn.(10) Tuan, should we define Fn? Let us recall the following relationship between the Mittag-Leffler functions Eα,1,Eα,α and the natural exponential function, see e.g. Section 2 in [1], Eα,1(−ρntα) = Z∞ 0 Φα(y) exp −ytαρndy, Eα,α(−ρntα) = αZ∞ 0 yΦα(y) exp −ytαρndy, where Φαdenotes the Wright type function introduced by Mainardi in [41] Φα(y) = ∞ X j=0 yn n!Γ(1 −α(1 + j)), y ∈C. This function is an entire function on C. The following classical result provides some essential relations used in this paper to obtain the main estimates. Proposition 2.1. For α∈(0,1) and θ > −1. Then the following properties hold: Φα(y)≥0,∀y≥0,and Z∞ 0 yθΦα(y)dy =Γ(θ+ 1) Γ(θα + 1),∀θ > −1.(11) This follows from (9) that un(t) = bα ωα ηnZt 0 R(t, τ)Fn(u(·, τ))dτ +Z∞ 0 Φα(y)e−ytαρnηnφndy +αµnZt 0Zt0 0Z∞ 0 yΦα(y)(t−t0)αR(t0, τ)e−y(t−t0)αρnFn(u(·, τ))dydτdt0,(12) where Φαis the Mainardi function or a particular Wright function, see e.g. [3, 4]. Now, let us define the following operators Bρϕ:= ∞ X j=1 ρnϕnbn,Bηϕ:= ∞ X j=1 ηnϕnbn,Bµϕ:= ∞ X j=1 µnϕnbn. Then, it follows from (12) that u(x, t) = bα ωαZt 0 R(t, τ)BηF(u(x, τ))dτ +Z∞ 0 Φα(y)e−ytαBρBηφ(x)dy +αZt 0Zt0 0Z∞ 0 yΦα(y)(t−t0)αR(t0, τ)e−y(t−t0)αBρBµF(u(x, τ))dydτdt0.(13) Definition 2.3. If a function uin Lp(0, T;Lq(Ω)) with some suitable numbers p≥1, q≥1 satisfies Equation (13), then it is called a mild solution of Problem (1)-(3). 4 3. Global existence on weighted Lebesgue space 3.1. Fractional power operators and the solution spaces. Firstly, we recall some literature on fractional power operators. Then, we present some useful functional spaces where the solution space will be mentioned. For each number s≥0, we define Xs(Ω) := (ϕ= ∞ X n=1 ϕnbn∈L2(Ω) : ∞ X n=1 ϕ2 nλs n<∞), ϕn=ZΩ ϕ(x)bn(x)dx. Let us denote by Hs(Ω) the Sobolev-Slobodecki space Ws,p(Ω) when p= 2, and by Hs 0(Ω) the closure of C∞ c(Ω) in Hs(Ω). Through out of this paper, Dis assumed to be smooth enough such that C∞ c(Ω) is dense in Hs(Ω) for 0 < s < 1 2. This guarantees Hs 0(Ω) = Hs(Ω). Moreover, it is well-known that Xs(Ω) =            Hs 0(Ω),for 0 ≤s < 1 2, H1/2 00 (Ω) $H1/2 0(Ω),for s=1 2, Hs 0(Ω),for 1 2< s ≤1, H1 0(Ω) ∩Hs(Ω),for 1 < s ≤2, where we denote by H1/2 00 (Ω) the Lions–Magenes space. Let X−s(Ω) be the duality of Xswhich corresponds to the dual inner product (·,·)−s,s. Then, the operator As:Xs(Ω) →X−svof fractional powers scan be defined by Asϕ:= P∞ n=1 βs n(ϕn, bn)−s,s ϕn,∀υ∈Xs.The above settings can be found in [5] (Section 3), [6] (Section 2), [7] (Section 2) and therein. In the next lemmas, we present some useful embeddings between the spaces mentioned above. Lemma 3.1. Let 0≤s≤s0≤2and let H−s(Ω) be the dual space of Hs 0(Ω). Then the following embeddings hold Xs(Ω) ,→L2(Ω) ,→X−s(Ω),(14) and Xs0(Ω) ,→Xs(Ω) ,→Hs(Ω) ,→L2(Ω) ,→H−s(Ω) ,→X−s(Ω) ,→X−s0(Ω).(15) Lemma 3.2. Given 1≤p, q < ∞,0≤s≤s0<∞and s0−N p0≥s−N p. Then, there holds that Ws0,p0(Ω) ,→Ws,p(Ω).(16) Let us denote the following sets by O+ 1:= (s;p) : s=N 2,1≤p < ∞,O+ 2:= (s;p):0≤s < N 2,1≤p≤2N N−2s, O+ 3:= (s;θ) : s > N 2, θ =s−N 2,O−:= (s;p) : −N 2< s ≤0, p ≥2N N−2s. As a consequence of the above lemma, we deduce that: Hs(Ω) ,→Lp(Ω) for (s, p)∈ O+ 1∪ O+ 2, and Hs(Ω) ,→C0,θ(Ω ∪∂Ω) for (s, p)∈ O+ 3. In contrast, Lp(Ω) ,→Hs(Ω) for (s;p)∈ O−. These combine with the chain (16) to allow the following lemma. Lemma 3.3. a) There hold that Xs(Ω) ,→Lp(Ω),(s;p)∈ O1∪O2,and Xs(Ω) ,→C0,θ(Ω∪∂Ω),(s;p)∈ O3.Moreover, Lp(Ω) ,→Xs(Ω),(s;p)∈ O−.(17) According to Definition 2.3, the solution space should be Lp(0, T;Lq(Ω)). In fact, since the purpose of the present paper is to investigate the global existence of mild solutions, it is necessary to narrow where we search for solutions. This is the rationale for introducing the so-called weight Lebesgue space Lp,γ,z(0, T ;Lq(Ω)). For given numbers p, q ≥1, γ > 0, z > 0, this is the space containing all functions ϕ∈Lp(0, T;Lq(Ω)) such that Lp,γ,z(0, T ;Lq(Ω)) = nϕ∈Lp(0, T;Lq(Ω)),ktγe−ztϕkLp(0,T ;Lq(Ω)) <∞o(18) with the corresponding norm kϕkLp,γ,z (0,T ;Lq(Ω)) := ktγe−ztϕkLp(0,T ;Lq(Ω)) (19) 5 3.2. Global existence assertion with globally Lipschitz nonlinearity. Our goal in this part is to establish the global existence of a mild solution of Problem (1)-(3). The following assumption will be needed throughout the paper: •(AS) N≥2, 1 ≤p < 2, and q≥2 such that N N−β< q ≤2N N−2β; •(AF) Let rbe satisfied 0 ≤r≤β−N 2−N q. Suppose that the function F:X0(Ω) →X−r(Ω) satisfying  F(h1)−F(h2) X−r(Ω) ≤KF h1−h2 X0(Ω),(20) for all h1, h2∈X0(Ω). Assume, furthermore, that F(0) = 0; •(AR) Assume that κ>1−p pand ν > 2p−2 p. Additionally, there exists a positive constant R0 such that |R(t, τ)| ≤ R0(t−τ)κτν,(21) for all 0 < τ < t < T. In the following theorem, we present a global existence of mild solutions to Problem (1)-(3). The word global indicates that there are no any restrictions on the time Tand the Lipschitz coefficient KF. Theorem 3.4. Given 0< α < 1,0< β ≤2. Assume that hypothesis (AS) holds. If φ∈LNq N+βq (Ω) and assumptions (AF), (AR) hold, then there exists z0>0such that Problem (1)-(3) has only one mild solution u∈Lp,γ,z(0, T ;Lq(Ω)),with 0< γ < νp +p−1. Proof. The proof is based on Banach contraction principle argument. Let us define the mapping S: Lp,γ,z(0, T ;Lq(Ω)) →Lp,γ,z(0, T ;Lq(Ω)) by Sv=S0φ+SF 1v+SF 2v, for all v∈Lp,γ,z(0, T ;Lq(Ω)), where the terms Sφand SFvare given by (S0φ)(x, t) := Z∞ 0 Φα(y)e−ytαBρBηφ(x)dy, SF 1v(x, t) = bα ωαZt 0 R(t, τ)BηF(u(x, τ))dτ, SF 2v(x, t) := αZt 0Zt0 0Z∞ 0 yΦα(y)(t−t0)αR(t0, τ)e−y(t−t0)αBρBµF(u(x, τ))dydτdt0.(22) Now we continue to split the proof into several steps. Step 1. Estimating the term S0φ Let us first estimate the quantity Bηφ(x). According to the definition (10), one has ωα+bαλβ/2 n=bα(1 + λβ/2 n) + α(Γ(α))−1≥bα(1 + λβ/2 n). In addition, for each 0 ≤ξ≤2, it holds that 1 + λβ/2 n≥λβξ/4 n. Therefore, we obtain   Bηφ   2 X Nq−2N 2q(Ω) = ∞ X n=1 η2 nλ Nq−2N 2q nφ2 n≤ ∞ X n=1 ωα ωα+bαλβ/2 n!2 λ Nq−2N 2q nφ2 n≤ω2 α bα2 ∞ X n=1 λ N 2−N q−βξ 2 nφ2 n, which respectively implies kBηφkX Nq−2N 2q(Ω) ≤ωα bαkφkX N 2−N q−βξ 2(Ω).(23) Moreover, for each number 0 ≤σ < 2, there exists a positive constant Cσsatisfying that e−ytαρn≤ Cσ(ytαρn)−σ/2for all nand all t > 0. We also note ρn=αλβ/2 n ωα+bαλβ/2 n ≥αλβ/2 1 ωα+bαλβ/2 1 =ρ1,as λn≥λ1. Henceforth, we obtain the following estimate   e−ytαBρBηφ   2 X N 2−N q(Ω) = ∞ X n=1 e−2ytαρnλ Nq−2N 2q nDBηφ, bnE2≤C2 σ ∞ X n=1 (ytαρn)−σλ Nq−2N 2q nDBηφ, bnE2, which implies that   e−ytαBρBηφ  X N 2−N q(Ω) ≤Cσρ−σ 2 1(ytα)−σ 2 Bηφ X N 2−N q(Ω).(24) From another point of view, it follows for q≥2 in assumption (AS) that the following Sobolev embedding XNq−2N 2q(Ω) ,→WN/2−N/q,2 x,→Lq(Ω),(25) 6 holds. Therefore, there exists M1>0 independently of xsuch that kϕkLq(Ω) ≤M1kϕkX Nq−2N 2q(Ω) for all ϕ∈XNq−2N 2q(Ω). This combines with the estimates (23) and (24) to allow the following chain of estimates   S0φ  Lp,γ,z (0,T ;Lq(Ω)) = ZT 0tγ pe−z pt S0φ Lq(Ω)pdt!1 p ≤ ZT 0tγ pe−z ptZ∞ 0 Φα(y) e−ytαBρBηφ Lq(Ω)dyp dt!1 p ≤M1 ZT 0tγ pe−z ptZ∞ 0 Φα(y) e−ytαBρBηφ X Nq−2N 2q(Ω)dyp dt!1 p ≤M1Cσρ−σ 2 1 ZT 0tγ p−ασ 2e−z ptZ∞ 0 y−σ 2Φα(y)dy Bηφ X Nq−2N 2q(Ω)p dt!1 p ≤M1 ZT 0 t(γ p−ασ 2)pe−ztdt!1 p φ X N 2−N q−βξ 2(Ω),(26) where M1=M1Cσρ−σ 2 1 Γ(1−σ 2) Γ(1−ασ 2) ωα bαand we have used the fact that (see (11)) R∞ 0y−σ 2Φα(y)dy =Γ(1−σ 2) Γ(1−ασ 2). Let us take ξ= 2. It is easy to see that since N N−β< q ≤2N N−2β, the pair N 2−N q−β;qbelongs to the set O−given in Lemma 3.3. Then, applying (17) of this lemma gives that the following Sobolev embedding LNq N+βq (Ω) ,→XN 2−N q−β(Ω),namely, there exists M2>0 independently of xsuch that kϕkX N 2−N q−β(Ω) ≤M2kϕkL Nq N+βq (Ω),(27) for any ϕ∈LNq N+βq (Ω). Moreover, by taking 0 ≤σ < min(2(γ+ 1)/(αp); 2), then the integral of the function t7→ t(γ p−ασ 2)pe−zt on (0, T) finitely exists. Consequently, we now imply from (26) that  S0φ Lp,γ,z (0,T ;Lq(Ω)) ≤M1M2 ZT 0 t(γ p−ασ 2)pe−ztdt!1 p φ L Nq N+βq (Ω),(28) which allows us to end the Step 1. Step 2. Estimating the term SF 2v In this step, an estimation for the term SF 2vwill be established. For the term SF 1v, we shall take it up in the next step by using essential techniques in this step. Firstly, we note that the embedding Lq(Ω) ,→L2(Ω) = X0holds as q≥2, i.e., k(v1−v2)(·, s)kL2(Ω) ≤M3k(v1−v2)(·, s)kLq(Ω), with a constant M3independent of x,s. On the other hand, by assumption 0 ≤r≤β−N 2−N qin (AF), we have N 2−N q−β≤ −r. Therefore, the following Sobolev embedding holds X−r(Ω) ,→XN 2−N q−β(Ω).(29) Then, there exists cF>0 such that kϕkX N 2−N q−β(Ω) ≤cFkϕkX−r(Ω) for all ϕ∈X−r(Ω). For the sake of simplicity let us denote e F(v1, v2) the difference F(v1)−F(v2). Then, by recalling the Sobolev embedding XNq−2N 2q(Ω) ,→WN/2−N/q,2 x,→Lq(Ω), 7 and using the analogue techniques as (23)-(24), we have   e−ytαBρBηe F(v1, v2)(·, τ)  Lq(Ω) ≤M1  e−ytαBρBηe F(v1, v2)(·, τ)  X Nq−2N 2q(Ω) ≤M1Cσρ−σ 2 1(ytα)−σ 2 Bηe F(v1, v2)(·, τ) X Nq−2N 2q(Ω) ≤M1Cσρ−σ 2 1ωα bα(ytα)−σ 2ke F(v1, v2)(·, τ)kX N 2−N q−β(Ω) ≤M1CσcFρ−σ 2 1ωα bα(ytα)−σ 2ke F(v1, v2)(·, τ)kX−r(Ω) ≤M1CσcFρ−σ 2 1ωα bαKF(ytα)−σ 2k(v1−v2)(·, τ)kL2(Ω) ≤M1M3CσcFρ−σ 2 1ωα bαKF(ytα)−σ 2k(v1−v2)(·, τ)kLq(Ω),(30) where the globally Lipschitz assumption (20) has been employed in the fourth estimate. Therefore, the difference SFv1− SFv2can be estimated as follows  SF 2v1− SF 2v2 Lp,γ,z (0,T ;Lq(Ω)) ≤α ZT 0 tγ pe−z ptZt 0Zt0 0Z∞ 0 (t−t0)αR(t0, τ) e−y(t−t0)αBρBµe F(v1, v2)(·, τ) Lq(Ω) (yΦα(y)) dydτdt0!p dt!1 p ≤M4 ZT 0 tγ pe−z ptZt 0Zt0 0Z∞ 0 (t−t0)(2−σ)α 2R(t0, τ) e F(v1, v2)(·, τ) XN 2−N q−βy2−σ 2Φα(y)dydτdt0!p dt!1 p ≤M5 ZT 0 tγ pe−z ptZt 0Zt0 0 (t−t0)(2−σ)α 2R(t0, τ) (v1−v2)(·, τ) Lq(Ω) Z∞ 0 y2−σ 2Φα(y)dydτdt0!p dt!1 p , (31) where the above constants are M4:= αM1M3CσcFρ−σ 2 1ωα bα, M5=M4KF.Now, it follows from H¨older’s inequality applied to the dual pair (p, p∗) (which means 1/p + 1/p∗= 1) that Zt 0Zt0 0 (t−t0)(2−σ)α 2R(t0, τ)k(v1−v2)(·, τ)kLq(Ω)dτdt0 =Zt 0 (t−t0)(2−σ)α 2Zt0 0 R(t0, τ)τ−γ pez pττγ pe−z pτk(v1−v2)(·, τ)kLq(Ω)dτdt0 ≤ Zt 0 (t−t0)(2−σ)α 2 Zt0 0 Rp∗ (t0, τ)τ−γp∗ pezp∗ pτdτ!1 p∗ dt0 kv1−v2kLp,γ,z (0,T ;Lq(Ω)) ≤R0 Zt 0 (t−t0)(2−σ)α 2 Zt0 0 (t0−τ)κp∗τνp∗−γp∗ pezp∗ pτdτ!1 p∗ dt0 kv1−v2kLp,γ,z (0,T ;Lq(Ω)).(32) Here, the integral on (0, t0) above can be calculated and then estimated by using the asymptotic behavior (8). In this integral, the parameters κp∗and νp∗−γp∗ pare strictly greater than −1. Indeed, one has κp∗=κp p−1>−1,as assumption κ>1−p pin (AR), and νp∗−γp∗ p=νp−γ p−1>νp−(νp+p−1) p−1=−1,as assumption 0 < γ < νp +p−1. Hence, it is easily checked that Zt0 0 (t0−τ)κp∗τνp∗−γp∗ pezp∗ pτdτ =M6(t0)νp∗−γp∗ p−1z−κp∗−1ezp∗ pt0t0+Op zp∗ ≤M7(t0)νp∗−γp∗ p−1z−κp∗−1ezp∗ pt01 + O1 z,(33) where we put M6= (p∗/p)−κp∗−1Γ(κp∗+ 1) and M7=M6max (T;p/p∗). Note that (2−σ)α 2>−1 as 0≤σ < 2, and ν−γ p−1 p∗=νp−γ−p+1 p>νp−(νp+p−1)−p+1 p>−1,as 0 < γ < νp +p−1, p < 2. It 8 follows that the following integral on (0, t) finitely exists. Summarily, we now obtain the following chain Zt 0Zt0 0 (t−t0)(2−σ)α 2R(t0, τ)k(v1−v2)(·, τ)kLq(Ω)dτdt0 ≤R0M 1 p∗ 7Zt 0 (t−t0)(2−σ)α 2(t0)ν−γ p−1 p∗ez pt0 dt01 + O1 z1 p∗ z−κ−1 p∗kv1−v2kLp,γ,z (0,T ;Lq(Ω)) ≤R0M8tν−γ p−1 p∗−1z−(2−σ)α 2−1ez ptt+Op z1 + O1 z1 p∗ z−κ−1 p∗kv1−v2kLp,γ,z (0,T ;Lq(Ω)) ≤R0M9tν−γ p−1 p∗−1z−(2−σ)α 2−1ez pt1 + O1 z1 + O1 z1 p∗ z−κ−1 p∗kv1−v2kLp,γ,z (0,T ;Lq(Ω)), where M8=M 1 p∗ 7(1/p)−(2−σ)α 2−1Γ(2−σ)α 2+ 1and M9=M8max(T;p). On the other hand, as an immediate consequence of the property (11) we have Z∞ 0 y2−σ 2Φα(y)dy =Γ2−σ 2+ 1 Γ2−σ 2α+ 1. Therefore, the above arguments accordingly imply that  SF 2v1− SF 2v2 Lp,γ,z (0,T ;Lq(Ω)) ≤Mσ1 + O1 z1 p∗ ZT 0tν−1 p∗−1z−(2−σ)α 2−1pdt!1 p z−κ−1 p∗kv1−v2kLp,γ,z (0,T ;Lq(Ω)) =Mσ1 + O1 z1 + O1 z1 p∗ ZT 0 tνp−p p∗−pdt!1 p z−(2−σ)α 2−1−κ−1 p∗kv1−v2kLp,γ,z (0,T ;Lq(Ω)). where Mσ=R0M5M9 Γ(2−σ 2+1) Γ(2−σ 2α+1)1 + O1 z. Here, the number νp −p p∗−pare strictly greater than −1 since ν > (2p−2)/p. Indeed, it is obvious to see that νp −p p∗ −p=νp + 1 −2p > 2p−2 pp+ 1 −2p=−1.(34) This ensures that the integral of t7→ tνp−p p∗−pon (0, T) finitely exists. Henceforth, we can conclude that there exists z1>0 large enough satisfying  SF 2v1− SF 2v2 Lp,γ,z (0,T ;Lq(Ω)) ≤1 4kv1−v2kLp,γ,z (0,T ;Lq(Ω)), for each given number z≥z1. Step 3. Estimating the term SF 1v Now, we also use notation e F(v1, v2) instead of F(v1)−F(v2). Recalling that the term Bηe F(v1, v2) may be proved in much the same way as (23) and (30). Thus, we have  SF 1v1− SF 1v2 Lp,γ,z (0,T ;Lq(Ω)) ≤M1bα ωα ZT 0tγ pe−z ptZt 0 R(t, τ) Bηe F(v1, v2)(·, τ) X N 2−N q(Ω)dτp dt!1 p ≤M1bα ωα ZT 0tγ pe−z ptZt 0 R(t, τ)ωα bαke F(v1, v2)(·, τ)kX N 2−N q−β(Ω)dτp dt!1 p . 9