EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION IN H σ,p 0

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EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE

LOGISTIC EQUATION IN H σ,p 0

J. Vanterler da C. Sousa, Mouffak Benchohra, Gastão Frederico

To cite this version:

J. Vanterler da C. Sousa, Mouffak Benchohra, Gastão Frederico. EXISTENCE, UNIQUENESS AND

REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION

IN H

EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION IN H^eσ,p₀

J. VANTERLER DA C. SOUSA, MOUFFAK BENCHOHRA AND GAST ˜AO S. F. FREDERICO

Abstract. In the present paper, we investigate the existence, uniqueness, regularity and continuous depen- dence of mild solutions to a fractional diffusive logistic equation with memory in Bessel potential spaces, through Banach fixed point theorem and Gronwall inequality.

1. Introduction

The logistic population model is considered as an important type of nonlinear differential equations due to ability to model several biological phenomena. The study of the logistical equation over the decades has been the subject of a study for many differential purpose [12, 14, 19, 20] and references therein. In the context of differential equation (integer order), it has also been considered, see for example [8,15,18] and references therein. Gopalsamy [17] had his research directed at the asymptotic behavior of non-constant solutions of delay logistic equations. Feng and Lu [18], also dedicated to investigating the asymptotic periodicity in diffusive logistic equations with discrete delay, i.e., as follows

∂u(t, x)

∂t −Au(t, x) =u(t, x)

"

a(t, x)−b(t, x)u(t, x)−

m

X

r=1

Cr(t, x)u(t−rT, x)

#

, [0,∞)×Ω B[u](t, x) = 0, [0,∞)×∂Ω

u(s, x) =u0(s, x), [−mT,0]×Ω.

Under certain conditions, the existence of solutions to the problem

∂u(t, x)

∂t = ∆u(t, x) +u(t, x)

a−bu− Z t

0

λ(t−s)u(s, x)ds

, [0,∞)×Ω

∂u

∂n = 0 u(x,0) =u₀(x)

for (t, x)∈(0,∞)×Ω, was proved by Schiaffino [22] and Yamada [23].

On the other hand, fractional derivatives provide excellent instrument for the description of memory.

In this sense, several approaches regarding the fractional logistic equation have been and over the years it has gained more prominence [9, 11]. In 2018, Ezz-Eldiem [10], discussed a numerical approach on solving fractional logistic population models via Jacobi polynomials. In 2020, Marinelli and Mugnai [7], discussed quantitative and geometric properties for the fractional generalized logistic equation given by

(−∆)^su(x) = λ(β(x)u(x)−g(x, u(x))), in Ω

u = 0, onR^N/Ω

when N > 2s, s ∈ (0,1), λ ∈ R, Ω ⊂ R^N is a bounded domain of class C^2α for some 0 < α ≤ 1 and g : Ω×R→R is a self-limiting factor for the populations. Other works on fractional logistic equation can be obtained at [9,11].

Motivated by the works above, in this paper, we consider the fractional logistic equation with memory effect from the initial data given by

(1.1) ^cD^α_tξ(t, x) = ∆ξ(t, x) +ξ(t, x)

a−b Z t

0

λ(t−s)ξ(s, x)ds

2010Mathematics Subject Classification. 26A33,35R11,35K20,35D30.

∗Correspondent author.

Key words and phrases. Fractional partial integrodifferential equation, existence and uniqueness, regularity, Bessel potential spaces, fractional logistic equation.

1

2 J. VANTERLER DA C. SOUSA, MOUFFAK BENCHOHRA AND GAST ˜AO S. F. FREDERICO

where ^cD^α_tξ(·) is the Caputo partial fractional derivative of order 0< α <1 , with ξ(t, x) is the population concentration inxandt >0,aandb are the growth rates and the agglomeration effect, respectively. Here, λ:R→Ris a function as a delay kernel representing the history of the species that influences the current growth rate.

We also consider a more general Cauchy-Dirichlet problem in the sense of the Caputo partial fractional derivative, given by

(1.2)











cD^α_tξ(t, x) = ∆ξ(t, x) +ξ(t, x)

a−b Z t

0

λ(t−s) (−∆)^ηeξ(s, x)ds

in (0,∞)×Ω

ξ = 0 in (0,∞)×Ω

ξ(x,0) = ξ0(x) in Ω

in a sufficiently regular domain Ω⊂Rⁿ. Note that, the Eq.(1.2) comes down to Eq.(1.1), whenηe= 0.

In order to define the fractional derivative, it is important to recall some facts about the theory of fractional calculus. Forα >0, define the functiongα:R→R[1]

(1.3) gα(t) :=





 1

Γ(α)t^α−1, t >0

0 , t≤0

where Γ(α) is Euler’s Gamma function.

Now, assume that T > 0. For a functionv ∈L¹((0, T),X), the Riemann-Liouville fractional integral of orderαofξis given by [1,2,3,4, 5]

I_t^αξ(t) :=gα∗ξ(t) = 1 Γ(α)

Z t 0

(t−s)^α−1ξ(s)ds, t∈[0, T].

Observe that the previous identification associates the properties of convolutions with the definitions of fractional integral operator. Hence, based of the definition of Riemann-Liouville fractional integral operator, we present the Caputo fractional differential operator.

Definition 1.1. [1, 2,3,4,5]Let α∈(0,1) andT >0. Considerv∈C([0, T],X)such that the convolution g1−α∗v∈W^1,1((0, T),X). The expression

cD^α_tξ(t) := d dt

I_t^1−α(ξ(t)−ξ(0)) = d dt

1 Γ(1−α)

Z t 0

(t−s)^−α(ξ(s)−ξ(0))ds

is called the Caputo fractional derivative of orderαof the functionξ(·).

Now, we consider the following conditions:

(H). We assume that

1< p <∞,σe∈(0,2)\ {1/p} andσe≥2β+ N 2p. Under these conditions, we use results contained in [24] to obtain the embedding (1.4) H^eσ,p₀ ,→ H^σ,pe (Ω),→ H^2β,2p(Ω),→L^2p(Ω),→L^p(Ω).

Inspired by above questions, we will use the Banach fixed point and Gronwall inequality to consider the existence, uniqueness, regularity and continuous dependence of mild solution for the fractional diffusive logistic equation Eq.(1.2) involving Caputo fractional derivative inH^eσ,p₀ , in other words, we investigate the following:

Theorem 1.2. Letλ:R→Rbe a locally integrable function and assume that(H) holds and(t1−s)^α−1− (t2−s)^α−1 ≤L(t2−t1)^α−1 with t2 > t1, 0 < α <1 and that exist (Eα(t^α∆))⁻¹. Given v0 ∈ H^eσ,p₀ , there exist τ >0 andr >0 such that for everyξ0∈B_Heσ,p

0

(v0, r)the fractional Cauchy-Dirichlet problem Eq.(1.2) possesses a unique mild solution ξ: [0, τ)→ H₀^eσ,p. Further, ξ∈C

(0, τ];H^σ₀^e1,p

forσe1∈[σ/2)\ {1/p}e and the solution continuous dependence on the initial data.

Our paper is organized as follows: Section 2, we present the Gronwall inequality and the definition of mild solution for Eq.(1.2). In this sense, we investigated an essential result for the development of this paper (see Lemma2.5). In section 3, we investigate the main results of this paper, i.e., the existence, uniqueness, regularity and continuous dependence of mild solution to Eq.(1.2) in Bessel potential spaceH^eσ,p₀ .

EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION INH₀3

2. Mathematical Background - Auxiliary Results Consider the initial data in the Bessel space [21]

(2.1) H₀^eσ,p=n

θ1∈ H^eσ,p(Ω) ;θ1|∂Ω= 0o with 1< p <∞e 0<σ <e 2.

Given a Banach spaceY, as usual,k·k_Ydenotes the norm associated toY. The ball of radius and centered at x∈Yis denoted byB_Y(x, r). IfXandYare Banach spaces, X,→Ymeans thatXis continuously and densely embedded inY.

LetS⊂RandXbe a Banach space. For 1≤p≤ ∞,L^p(S,X) denotes the Banach space ofL^pintegrable functionsv:S→Xifp <∞and the essential bounded functions whenp=∞. W^1,p(S,X) is the subspace of L^p(S,X) consisting of functions such that the weak derivativevtbelongs toL^p(S,X). Both spacesL^p(S,X) andW^1,p(S,X) are endowed with their standard norm.

Let αand β be strictly positive real numbers. Then, Eα,β(z) : C→ Cis the Mittag-Leffler function of two-parameters, given by [1]

(2.2) Eα,β(z) =

∞

X

k=0

z^k Γ (αk+β).

Choosingβ = 1, we have the Mittag-Leffler function of one-parameter, given by

(2.3) Eα(z) =

∞

X

k=0

z^k Γ (αk+ 1).

Theorem 2.1. [6] (Gronwall inequality)Let u, v be two integrable functions and g a continuous function, with domain [0, T]. Letψ∈C¹[0, T]be an increasing function such thatψ⁰(t)6= 0,∀t∈[0, T]. Assume that functionsuandv are nonnegative and g in nonnegative and nondecreasing.

If

(2.4) u(t)≤v(t) +g(t)

Z t 0

ψ⁰(τ) (ψ(t)−ψ(τ))^α−1dτ t∈[0, T] andv be a nondecreasing function on[0, T], then

(2.5) u(t)≤v(t)Eα(g(t) Γ (α) [ψ(T)−ψ(0)]^α)

∀t∈[a, b], whereEα(·)is the Mittag-Leffler function given byEq.(2.3).

Definition 2.2. A continuous functionξ: [0, τ]→ H^eσ,p₀ is said to be a mild solution for Eq.(1.2), if it is a solution of the following integral equation

(2.6) ξ(t) =Eα(∆t^α)ξ₀+ Z t

0

(t−s)^α−1Eα,α((t−s)^α∆)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^eηξ(r)dr

ds.

Let 1< p <∞ and letσe∈(0,2)\ {1/p}. Note that, the Bessel potential spaceH^σ,p₀^e coincides with the complex interpolation space

W^2,p∩W^1,p, L^p(Ω)

eσ/2 for 0<eσ <2,eσ6= 1/p.

It is well known that the Dirichlet Laplacian ∆ is a sectorial operator from W^2,p∩W₀^1,p into L^p(Ω).

Therefore, the operatorEα(∆t^α) :L^p(Ω)→L^p(Ω) satisfies the following estimative [12,13]

(2.7) t^σf21−eσ2 kEα(∆t^α)θ1k_Hσ,pe 0

≤Lkθ1k_Heσ,p 0

for all θ1 ∈ H₀^eσ,p and t >0 , where L ≥1. Here 0≤ eσ≤ eσ1 <2 and neither of eσ,σe1 is equal to 1/p. In particular, ifσe∈(0,2)\ {1/p}, then

(2.8) t^αe2σ kEα(∆t^α)θ₁k_Hσ,pe 0

≤Lkθ₁k_L

p

for allθ1∈Lp(Ω) and t >0.

Remark 2.3. [16]The following estimate is essential to treat the term which involves the fractional Laplacian

(2.9)

(−∆)^eσ/2θ_{1 L}

p

≤Lkθ1k_Heσ,p

4 J. VANTERLER DA C. SOUSA, MOUFFAK BENCHOHRA AND GAST ˜AO S. F. FREDERICO

for allθ₁∈ H^σ,pe .This is a consequence of the equivalence between the norms

(−∆)^σe(·) _L

p

and

I−(−∆)^σe (·)

_L

p

the fact the

I−(−∆)^eσ (·)

_L

p(R^N)is a norm onH^2eσ,p R^N

, and that the extension operatorE:H^s,p(Ω)→ H^s,p R^N

is continuous.

Lemma 2.4. [16] Let λ:R→R be a locally integrable function. Assume that(H)holds and consider the function g: [0,∞)× H^σ,p₀^e →L^p defined by

g(t, θ1) =θ1

a−b

Z t 0

λ(t−s) (−∆)^βθ1ds

. Then, given θ1, θ2∈ H^eσ,p₀ , there existsc >0 such that

kg(t, θ1)−g(t, θ2)k_Lp(Ω)≤ckθ1−θ2k_Hσ,pe 0

nkθ1k_Hσ,pe

0 +kθ2k_Hσ,pe 0

+ 1o and

kg(t, θ1)k_Lp(Ω)≤ckθ1k_Hσ,pe 0

nkλk_L1(0,t)+kθ1k_Heσ,p 0

+ 1o

where the constant c depends ona, b,[Ω] (the Lebesgue volume ofΩ) and the embedding(1.4).

Lemma 2.5. Letλ:R→Rbe a locally integrable function. Assume that (H) holds and consider functions ξi: [0, τ]→ H^eσ,p₀ such that

(2.10) sup

t∈[0,τ]

kξi(t)k_Hσ,pe 0

≤µ⁰,i= 1,2

whereµ⁰>0and we suppose that λ: [0,∞)→[0,∞) is locally integrable. Then, we have

Z t 0

Eα,α(∆(t−s)^α) (g(r, ξ1(r))−g(r, ξ2(r)))drds _Hσ,pe

0

≤ Lc sup

s∈[0,t]

kξ1(s)−ξ2(s)k_Hσ,pe 0

2kλk_L1

loc(0,t)µ⁰+ 12t^1−σe2 2−eσ and

Z t 0

Eα,α(∆ (t−s)^α)g(r, ξ1(r))drds _Hσ,pe

0

≤Lcµ⁰ 2kλk_L1

loc(0,t)µ⁰+ 12t^1−eσ2 2−σe. Proof. Indeed, using the Remark2.3and Lemma2.4, yields

Z t 0

Eα,α(∆(t−s)^α) (g(r, ξ1(r))−g(r, ξ2(r)))drds _Hσ,pe

0

≤ Z t

0

L(t−s)^−αe2σkg(r, ξ₁(r))−g(r, u₂(r))k_Lp(Ω)drds

≤ Lc Z t

0

(t−s)^−αe2σ kξ1(r)−ξ2(r)k_Hσ,pe 0

hkλk_L1 loc(0,t)

kξ1(r)k_Hσ,pe

0 +kξ2(r)k_Heσ,p 0

+ 1i

drds

≤ Lc sup

s∈[0,t]

kξ₁(r)−ξ₂(r)k_Hσ,pe 0

2kλk_L1

loc(0,t)µ⁰+ 12t^1−eσα2 2−eσα. (2.11)

On the other hand, we have

Z t 0

Eα,α(∆ (t−s)^α)g(r, ξ(r))drds H^σ,p₀^e

≤ L Z t

0

(t−s)^−αe2σkg(r, ξ(r))k_Lp(Ω)drds

≤ Lcµ⁰ 2kλk_L1

loc(0,t)µ⁰+ 12t^1−eσα2 2−eσα. (2.12)

EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION INH₀5

3. Proof of Theorem1.2

In this section, we discuss the main results of this paper, that is, the existence, uniqueness, regularity and continuous dependence of mild solutions of type Eq.(2.6) in Bessel potential space H^σ,p₀^e .

Proof. Let 0< µ≤1 and 0< α <1. Chooseτ >0 small enough so that, for allt∈[0, τ],

(3.1) kEα(∆t^α)v0−v0k_Hσ,pe

0

≤ µ 3 and

(3.2) Lc

kλk_L1(0,t)µ⁰+ 1 µ⁰ 2

2α−eσαt^α−αeσ/2<µ 3 where µ⁰:=µ+kv0k_Hσ,pe

0

. Setr= µ

3L. In this sense, yields

(3.3) kEα(∆t^α)ξ0−v0k_Hσ,pe

0

≤2µ 3 . Now, we define

(3.4) Be=

( ξ∈C

[0, τ] ;H^eσ,p₀ , sup

t∈[0,t]

kξ(t)−v0k_Hσ,pe 0

≤µ )

. For proving these results, we consider the map Λ :Be→Be defined by

(3.5) (Λξ) (t) =Eα(∆t^α)ξ₀+ Z t

0

(t−s)^α−1Eα,α(∆ (t−s)^α)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^ηeξ(r)dr

ds.

First, let us prove that Λ is well defined, that is, ΛBe⊂B. Hence, for 0e ≤t₁< t₂≤τ, we have k(Λξ) (t1)−(Λξ) (t2)k_Heσ,p

0

≤ Z t1

0

h(t₁−s)^α−1−(t₂−s)^α−1i

[Eα,α(∆ (t₁−s)^α)−Eα,α(∆ (t₂−s)^α)]

_Hσ,pe

0

×

u(s)

a−b Z s

0

λ(s−r) (−∆)^ηeu(r)dr

_Heσ,p

0

ds+kEα(∆t^α₁)ξ0−Eα(∆t^α₂)ξ0k_Heσ,p 0

+ Z t2

t1

(t2−s)^α−1Eα,α(∆ (t2−s)^α)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^ηeξ(r)dr

H^eσ,p₀

ds

≤ kEα(∆t^α₁)ξ₀−Eα(∆t^α₂)ξ₀k_Hσ,pe 0

+

I−Eα(∆ (t₁−τ)^α) (Eα(∆ (t₂−τ)^α))⁻¹ _L(^Heσ,p0 )

×L Z t

0

h

(t1−s)^α−1−(t2−s)^α−1i

(t2−s)^−αeσ/2

ξ(s)

a−b Z s

0

λ(s−r) (−∆)^eηξ(r)dr

_Lp(Ω) +

Z t2

t₁

(t₂−s)^{α−αe2σ−1}L

ξ(s)

a−b Z s

0

λ(s−r) (−∆)^ηeξ(r)dr

_Lp(Ω)

ds

≤ kE^α(∆t^α₁)ξ0−E^α(∆t^α₂)ξ0k_Hσ,pe 0

+

I−E^α(∆ (t1−τ)^α) (E^α(∆ (t2−τ)^α))⁻¹ _L(^Heσ,p0 )

×Lcµ⁰

kλk_L1(0,t₁)µ⁰+ 1Z t₁ 0

h(t₁−s)^α−1−(t₂−s)^α−1i

(t₂−s)^−αeσ/2ds +Lcµ⁰

kλk_L1(0,t1)µ⁰+ 1Z t2

t1

(t₂−s)^{α−αe2σ−1}ds

≤ kEα(∆t^α₁)ξ₀−Eα(∆t^α₂)ξ₀k_Hσ,pe 0

+

I−Eα(−∆ (t1−τ)^α) (Eα(−∆ (t2−τ)^α))⁻¹ _L(^Hσ,p0^e )

×L²cµ⁰

kλk_L1(0,t₁)µ⁰+ 1

(t2−t1)^α

(2 (t2−t1)^1−αe2σ

2−αeσ −2t21−^αe₂^σ

2−ασe )

+cLµ⁰

kλk_L1(0,t1)µ⁰+ 12 (t2−t1)

α(2−eσ) 2

α(2−σ)e (3.6)

6 J. VANTERLER DA C. SOUSA, MOUFFAK BENCHOHRA AND GAST ˜AO S. F. FREDERICO

which converges to zero as either t₁→t⁻₂ or t₁→t⁺₁. Using the Lemma2.5, yields k(Λξ) (t)−v₀k_Hσ,pe

0

≤ kEα(∆t^α)ξ₀−v₀k_Hσ,pe 0

+ Z t

0

(t−s)^α−1

×

Eα,α(∆ (t−s)^α)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^eηξ(r)dr

H^σ,p₀^e

ds

≤ µ

3 +Lcµ⁰

kλk_L1(0,t₁)µ⁰+ 1Z t 0

(t−s)^{α−αe2σ−1}ds

< µ 3 +2µ

3 =µ.

Hence, we concluded that, Λ is well defined.

Next, we show that Λ is a contraction. Foru, v∈B, by Lemmae 2.5, yields k(Λξ) (t)−(Λv) (t)k_Heσ,p

0

≤ Z t

0

(t−s)^α−1

Eα,α(∆ (t−s)^α) (ξ(s)−v(s))

a−b Z s

0

λ(s−r) (−∆)^ηe(ξ(r)−v(r))dr

_Hσ,pe

0

ds

≤ Lc2τ^α−αeσ/2 2α−αeσ

2kλk_L1(0,t₁)µ⁰+ 1 sup

s∈[0,t]

kξ(s)−v(s)k_Hσ,pe 0

≤ 1 3 sup

s∈[0,t]

kξ(s)−v(s)k_Heσ,p 0

.

Using the Banach fixed point theorem, Λ has a unique fixed pointξ∈B. This is a mild solution for (1.2).e If we repeat these steps, but using (2.7) instead of (2.8) and using the same reasoning as (3.6), we obtain

kξ(t₁)−ξ(t₂)k

H^eσ₀^f1,p

≤

Z t1

0

(t₁−s)^α−1Eα,α(∆ (t₁−s)^α)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^ηeξ(r)dr

ds H^σ,p₀^e

+

Z t₂ 0

(t2−s)^α−1Eα,α(∆ (t2−s)^α)ξ(s)

a−b Z s

0

λ(s−r) (−∆)^ηeξ(r)dr

ds _Hσ,pe

0

+kEα(∆t^α₁)ξ0−Eα(∆t^α₂)ξ0k_Hσ,pe 0

≤

I−Eα(∆ (t1−τ)^α) (Eα(∆ (t2−τ)^α))⁻¹ _L

H^σ₀^e1,pL²cµ⁰

kλk_L1(0,t₁)µ⁰+ 1

×(t₂−t₁)^α







2 (t₂−t₁)¹⁻

αeσ1 2

2−αeσ1

−2t¹⁻

αeσ1 2

2

2−αeσ1







+Lcµ⁰

kλk_L1(0,t₁)µ⁰+ 12 (t₂−t₁)

α(2−eσ1 ) 2

α(2−σe1) +

I−Eα(∆t^α₁) (Eα(∆t^α₂))⁻¹ _L

H^σ₀^e1,pkEα(∆t^α₂)ξ0k

≤

I−Eα(∆ (t₁−τ)^α) (Eα(∆ (t₂−τ)^α))⁻¹ _LHσe1,p

0

L²cµ⁰

kλk_L1(0,t1)µ⁰+ 1

×(t2−t1)^α







2 (t2−t1)¹⁻

αeσ1 2

2−αeσ₁ −2t¹⁻

αeσ1 2

2

2−αeσ₁





 +

I−Eα(∆t^α₁) (Eα(∆t^α₂))⁻¹ _LHσe1,p

0

Lt^{αe2σ−αeσ21} ku0k_Heσ,p 0

(3.7)

0 < t1 < t2 ≤ τ. Consequently, ξ ∈ C

[0, τ] ;H₀^eσ1,p

. This shows the existence and regularity of mild solution.

To finish the proof, we prove the uniqueness and consequently the continuous dependence on the data of the mild solution (see Eq.(2.6)). Letξebe a mild solution of (1.2). Then, using the Lemma2.4, yields

ξ(t)−ξe(t) _Hσ,pe

0

EXISTENCE, UNIQUENESS AND REGULARITY OF MILD SOLUTION TO A FRACTIONAL DIFFUSIVE LOGISTIC EQUATION INH₀7

≤ Z t

0

(t−s)^α−1

Eα,α(∆ (t−s)^α)

ξ(s)−ξe(s) a−b

Z s 0

λ(s−r) (−∆)^ηe

ξ(r)−ξe(r) dr

H^eσ,p₀

ds

≤ L Z t

0

(t−s)^{α−1−αe2σ}

g(ξ(r))−g ξe(r)

H^eσ,p₀ drds

≤ Lc

2kλk_L1(0,t)η+ 1Z t 0

(t−s)^{α−1−αe2σ} sup

r∈[0,s]

ξ(r)−ξe(r) _Hσ,pe

0

ds (3.8)

where η:= max (

sup

t∈[0,τ]

kξ(t)k_Hσ,pe 0

, sup

t∈[0,τ]

ξe(t)

H^σ,p₀^e

) . Putf(t) := sup

s∈[0,t]

ξ(s)−ξe(s) _Hσ,pe

0

andCe=Lc

2kλk_L1(0,t)η+ 1

,it follows that

(3.9) f(t)≤Ce

Z t 0

(t−s)^{α−1−αe2σ} f(s)ds

for all t ∈ [0, τ]. Using the Gronwall inequality (Theorem2.1), f(t) = 0 for all t ∈ [0, τ] and uniqueness follows.

Finally, taking ξ₁, ξ₂ ∈ B

Hσ,pe 0

(v₀, r) and for i = 1,2, let ξ_i(t) be the mild solution that starts at ξ_i, i= 1,2, .Then, using the same reasoning of inequality (3.8), we have

ξ(t)−ξe(t)

H^σ,p₀^e ≤ kEα(∆t^α)ξ1−Eα(∆t^α)ξ2k_Hσ,pe 0

+Lc

2kλk_L1(0,t)µ⁰+ 1Z t 0

(t−s)^{α−1−αe2σ} sup

r∈[0,s]

kξ1(r)−ξ₂(r)k_Heσ,p 0

ds

≤ kEα(∆t^α)ξ1−Eα(∆t^α)ξ2k_Hσ,pe 0

+Lc

2kλk_L1(0,t)µ⁰+ 1 sup

s∈[0,t]

kξ1(s)−ξ₂(s)k_Heσ,p 0

2τ^α−αe2σ 2α−αeσ

≤ kξ₁−ξ₂k_Hσ,pe 0

+ µ 2µ⁰ sup

s∈[0,t]

kξ₁(s)−ξ₂(s)k_Heσ,p 0

. (3.10)

Hence,

(3.11) sup

s∈[0,t]

kξ1(s)−ξ₂(s)k_Hσ,pe

0 ≤3

2kξ1−ξ₂k_Hσ,pe 0

.

Thus the mild solution of Eq.(1.2) is continuous dependence on the initial data. In this sense, we concluded

the proof.

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(J. Vanterler da C. Sousa)Center for Mathematics, Computing and Cognition, Federal University of ABC, Avenida dos Estados, 5001, Bairro Bangu, 09.210-580, Santo Andr´e, SP - Brazil

Email address:[email protected],[email protected]

(Mouffak Benchohra) Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P. O. Box 89, 22000, Sidi Bel-Abbes, Algeria

Email address:[email protected]

(Gast˜ao S. F. Frederico)Federal University of Cear´a, Campus de Russas, Russas, 62900-000 Brazil Email address:[email protected]