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1 Introduction

In recent years, the research on the time-space fractional diffusion equation with fractional Laplacian has attracted wide attention of scholars. The time-space fractional diffusion equation, which is generalizations of classical diffusion equation of integer order, is one of the most commonly used models to describe several anomalous physical aspects and procedures in natural conditions, such as mechanics of materials, fluid mechanics, image processing, finance, biology, signal processing and control (see [5, 10, 21, 22, 26,27,28]). The initial value problems for the time-space fractional diffusion equation have been extensively studied, and many properties of their solutions have been studied because of the importance in applications (see [3, 11, 18,19,20] and references therein).

Recently, the initial boundary value problems of the time-space fractional diffusion equations with fractional Laplacian have been considered by several authors (see [9, 12, 24, 29]). In [9], Chen et al. studied a homogeneous time-space diffusion equation. Combining the Mittag–Leffler function to the time fractional problem with an eigenfunction expansion of the fractional Laplacian on bounded domains, the existence of strong solutions was obtained by separation of variables. In [12], Jia and Li focused on an inhomogeneous time-space fractional diffusion equation. By utilizing properties of time fractional derivative operator and fractional Laplace operator, maximum principles for classical solution and weak solution were obtained. In [29], Toniazzi focused on an inhomogeneous time-space fractional linear diffusion equation involving time nonlocal initial condition and proved the existence and uniqueness of classical solutions along with the stochastic representation for the solution. In all these works, the existence theory of solutions for the semilinear equations is not involved.

Specially, Padgett in [24] investigated the initial boundary value problem for the timespace fractional semilinear diffusion equation with fractional Laplacian

where Ω is a bounded open domain in Rd with sooth boundary ∂Ω, cDα denotes the Caputo time-fractional derivative of order α ϵ (0, 1), and ( - ∆)β is the fractional Laplacian with β ϵ (0, 1). Under the assumption that the nonlinear reaction term f satisfies a local Lipschitz condition, the author has obtained the existence and uniqueness of (1) by means of Banach fixed point theorem. In fact, in the complex reaction-diffusion processes, the nonlinear function f represents the source of material or population, which depends on time in diversified manners in many contexts. Thus, we hope that the nonlinear function f satisfies more general growth conditions than Lipschitz type conditions.

On the other hand, the monotone iteration technique for upper and lower solutions is an effective and widely used mathematical method. By using this method not only the existence theory of solutions can be obtained, but also the approximate iteration sequence of solutions can be obtained, which provides a reasonable and effective theoretical basis for using the computer to obtain the approximate solution. However, as far as we know, there are few results for the diffusion equations with delay by means of the method for the lower and upper solutions coupled with the monotone iterative technique (see [15, 16]).

Motivated by the papers mentioned above, we study the following initial boundary value problem for the fractional delayed semilinear diffusion equation with fractional Laplacian:

where Ω ∈ Rd is a bounded domain with C2-boundary ∂Ω for d ∈ N, 0 < α, β ≤ 1, cDα denotes the Caputo fractional derivation of order α ∈ (0, 1), (−∆)β is a realization of the fractional Laplace operator acting in space. f : Ω × [0, a] × R2 → R is continuous functions, ϕ ϵ C(R X [ - r, 0]), r > 0, is a constant. In this paper, our main purpose is to establish a general principle of lower and upper solutions coupled with the monotone iterative technique to the initial boundary value problem (2) and study the existence of maximal and minimal mild solutions, which will greatly enrich and expand the results mentioned above.

As we all know, there are many different definitions of Laplacian operator and fractional Laplacian operator on a bounded domain Ω. For the properties of differential definitions and their relations, we can refer to [14, 17, 21] and references therein. Once the Laplace operator ∆ is defined, according to Balakrishnan’s definition, a common definition of fractional Laplacian is provided by fractional power of the nonnegative operator −∆ (see [4, 32])

for u ϵ D( ∆)—the domain of the consider Laplace operator. Throughout this paper, we introduce the definition of function calculus of fractional Laplacian through Dirichlet Laplacian, which means that - ∆ : L2(Ω) L2(Ω) is the classical Laplacian with domain D( - ∆) = {u H1(Ω) ∆u L2(Ω) } . As we all know, the operator is unbounded, closed, positive define self-adjoint and has a compact inverse.

Hence, if λi (i = 1, 2 . . . ) are the eigenvalues of ∆ with homogeneous Dirichlet boundary conditions considered in L2(Ω) and ei as its corresponding eigenfunction, then

Thus, we can define the fractional Laplacian to be

From [6, 7] it follows that the Balakrishnan definition is equivalent to the spectral definition in L2(Ω).

The structure of this paper is as follows. In Section 2, we collect some known concepts and results about the operator semigroup and provide preliminary results, which can be used in the theorems stated and proved in this paper. In Section 3, we present our abstract results and apply the operator semigroup theory and monotone iterative method of the lower and upper solution to prove them. In the last section, applying our abstract results to the initial boundary value problem for the time-space fractional delayed semilinear diffusion equation with fractional Laplacian, we get the existence and uniqueness of positive solutions.

2 Preliminaries

Throughout this paper, we assume that (E, ǁ ǁ) is an ordered Banach space with the partial-order “≤” induced by the positive cone K = {u ϵ E │ u ≥ θ } and K is normal, θ is the zero element of E.

Denote J │:= [−r, a], and let C(J, E) be the Banach space composed of all continuous functions from J to E equipped with the norm ǁuǁ C = maxt∈J ǁu(t) ǁ . Evidently, C(J, E) is also ordered Banach space, the positive cone KC = {u ϵ C(J, E) │ u(t) ϵ K, t ϵ J} is also normal. Similarly, β is ordered Banach space with norm ǁφǁ B = maxs ϵ [ - r,0] ǁφ(s)ǁ , and the positive cone KB = {φ ϵ mod φ(s) ϵ K, s ϵ [- r, 0]} .

For v, w ϵ C(J, E) with v ≤ w, we denote the order interval {u │ v ≤ u ≤ w} by [v, w]. Moreover, we denote {u(t) ≤ v(t) ≤ u(t) ≤ w(t), t ϵ J} in E and ut ≤ vt ≤ ut ≤ wt, t ϵ [0, a] in β by [v(t), w(t)] and [vt, wt], respectively.

Next, we recall some essential properties of operator semigroup.

Let A : D(A) E → E be a closed linear operator and - A generate a uniformly bounded C0-semigroup T (t) (t ≥ 0) on E. Thus, there is M ≥ 1 such that

From [25, 32] it follows that A is a nonnegative operator and

Therefore, for any 0 < β < 1, according to the Balakrishnan definition [4, 32], we can define the fractional power Aβ of the nonnegative operator A by

Then, from [32] we find that -Aβ is a closed densely defined operator and generates an analytic semigroup Tβ(t) (t ≥ 0), which can be expressed as

where fβ,t(·) is defined by

and the brach of zβ is so taken that Re(zβ) > 0 for Re(z) > 0. The convergence of integral (4) is apparent in virtue of the convergence factor e−tzβ. Moreover, fβ,t(s) ≥ 0 for all s > 0, and ∫0 ∞ fβ,t(s) ds = 1. For more properties of the function fβ,t(s), one can refer to [32].

By the definition of the semigroup Tβ(t) and the properties of fβ,t(s) one can find that Tβ(t) is continuous by operator norm and ǁTβ(t) ǁ ≤ M for any t ≥ 0 and β (0, 1). Moreover, in ordered Banach space E, if the C0-semigroup T (t) (t ≥ 0) generated by −A is positive, then the semigroup Tβ(t) (t ≥ 0) generated by −Aβ is positive for any β ∈ (0, 1). Furthermore, we can obtain the following lemma.

Lemma 1. If the uniformly bounded C0-semigroup T (t) (t ≥ 0) generated by −A is compact, then the semigroup Tβ(t) (t ≥ 0) generated by −Aβ is compact.

Proof. Let ε > 0 be arbitrary, and let

One can easily obtain that ∫ϵ∞ fβ,t(s)T (s - ε) ds is a linear bounded operator for every t > 0. Hence, by the compactness of the semigroup T (t) (t ≥ 0), Tβ,ε(t) is compact for every t > 0. On the other hand, note that

Hence, by the boundedness of fβ,t(s) in s and the compactness of Tβ,ε(t) for t > 0 one can obtain that Tβ(t) (t ≥ 0) is compact.

For more details of the definitions and properties of C0-semigroups or positive C0semigroups, see [23, 25, 32].

As for the definition of Caputo fractional derivation, we can refer to many references (see [8, 13, 30] and so on), which will not be repeated here. In the following, we only give some operators needed in this paper and their related properties.

For a given C0-semigroup T (t) (t ≥ 0), we define the family of operators Uα(t) (t ≥ 0) and Vα(t) (t ≥ 0) in E as follows:

Where

is a probability density function defined on (0, ∞), which satisfies

The following lemma may be find in [8, 30].

Lemma 2. The operators Uα(t) (t ≥ 0) and Vα(t) (t ≥ 0) have the following properties:

(i) Uα(t) (t ≥ 0) and Vα(t) (t ≥ 0) are strongly continuous operators, i.e., for any

x ∈ E and 0 ≤ t1 ≤ t2,

(ii) If C0-semigroup T (t) (t ≥ 0) is uniformly bounded, then Uα(t) and Vα(t) are linear bounded operators for any fixed t ∈ R+, i.e.,

(iii) If C0-semigroup T (t) (t ≥ 0) is compact, then Uα(t) and Vα(t) are compact operators for every t > 0.

(iv) If C0-semigroup T (t) (t ≥ 0) is continuous by operator norm for every t > 0, then Uα(t) and Vα(t) are uniformly continuous for t > 0.

(v) If C0-semigroup T (t) (t ≥ 0) is positive, then Uα(t) and Vα(t) are positive operators.

In the proof, we also need the following inequality.

Lemma 3. (See [31].) Assume that f (t) is a locally integrable, nonnegative function on 0 ≤ t < κ (some κ ≤ ∞ ), g(t) is a nonnegative, nondecreasing, continuous bounded function on 0 ≤ t < κ, and α > 0. Suppose that h(t) is locally integrable and nonnegative on 0 ≤ t < Λ with

Then

3 Abstract results

In this section, we discuss the existence of the minimum and maximum mild solutions for the abstract time-space fractional evolution equation with delay

where 0 < α, β ≤ 1, cDtα is the Caputo fractional derivation of order α ∈ (0, 1); A : D(A) ⊂ E → E is a closed linear operator, and −A generates a positive compact

semigroup T (t) (t ≥ 0) in E, which is uniformly bounded with supt≥0 ǁT (t)ǁ= M < ∞, Aβ denotes the βth fractional power operator of A according to the Blakrishman definition; F : [0, a] x E x β → E is a continuous function, which will be specified later; := C([ r, 0], E) denotes the space of continuous functions from [ - r, 0] into E provided with the uniform norm topology, r > 0 is a constant; ϕ ϵ β is given. For t ≥ 0, ut ϵ β denotes the history function defined by ut(s) = u(t + s) for s ϵ [ - r, 0], where u is a continuous function from [ -r, a] into E.

In order to introduce the definitions of the lower or upper solution and the mild solution for the time-space fractional delayed evolution equation (6), we set

and denote by E1 the Banach space D(A) with the graph norm ǁ·ǁ1 = ǁ·ǁ + ǁA · ǁ.

Definition 1. A function w ∈ C([−r, a], E) is said to be an upper solution of Eq. (6) if w|[0,a] ∈ Cα([0, a], E) ∩ C([0, a], E1) and

Definition 2. A function u ϵ C([ r, a], E) is said to be a mild solution of Eq. (6) if it satisfies

Next, we present and demonstrate our main results.

Theorem 1. Assume that Eq. (6) has upper and lower solutions w(0), v(0) satisfying v(0) ≤ w(0). If F : [0, a] × E × B → E is a continuous function satisfying

then Eq. (1) has maximal and minimal mild solutions u, u ∈ [v(0), w(0)].

Proof. Obviously, Eq. (6) can be rewritten in the following form:

where constant C is decided by condition (H1).

According to the discussion in the preparatory part, one can obtain that for any β ∈ (0, 1), −Aβ generated a uniformly bounded, positive and compact semigroup Tβ(t) (t ≥ 0) satisfying ǁTβ(t)ǁ ≤ M for every t ≥ 0.

We denote by Sβ(t) =e−CtTβ(t) (t ≥ 0) the C0-semigroup generated by (CI+Aβ).

Obviously,

Moreover, by the positivity and compactness of the semigroup Tβ(t) (t ≥ 0) one can see that Sβ(t) (t ≥ 0) is a positive compact semigroup. Define two operators Uα,β(t) (t ≥ 0) and Vα,β(t) (t ≥ 0) by

where x ∈ E, and ξα(s) is the function defined by (5). Thus, the operators Uα,β(t) (t ≥ 0) and Vα,β(t) (t ≥ 0) have properties (i)–(v) in Lemma 2, and for each t ≥ 0,

For each u ∈ [v(0), w(0)] and t ∈ [0, a], we have ut ∈ [vt(0), wt(0)] ⊂ B. Now, we define operator Q on [v(0), w(0)] by

From the normality of the cone K, condition (H1) and the continuity of F one can deduce that for any u ∈ [v(0), w(0)], there is a constant M0 > 0 such that

Hence, it is easy to show that Q : [v(0), w(0)] → C([−r, a], E) is well defined. By Definition 2 and (8) it can be asserted that u ∈ [v(0), w(0)] is a mild solution of Eq. (6) if u is a fixed point of Q.

Next, we prove it in four steps.

Step 1. Q : [v(0), w(0)] → [v(0), w(0)] is monotone increasing.

On the one hand, let

By the positivity of operators Uα,β(t) and Vα,β(t), for t ≥ 0, one can obtain that

and v(0)(t) ≤ ϕ(t) for t ϵ [ -r, 0]. Thus, v(0) ≤ v(0). Similarly, Qw(0) ≤ w(0) can be obtained.

On the other hand, for any u(1), u(2) ∈ [v(0), w(0)] with u(1) ≤ u(2) and t ≥ 0, we can see v(0)(t) ≤ u(1)(t) ≤ u(2)(t) ≤ w(0)(t), vt(0) ≤ u(1) ≤ ut(2) ≤ w(0). Thus, by condition (H1), F (t, u(2)(t), ut(2)) + Cu(2)(t) ≥ F (t, u(1)(t), ut(1)) + Cu(1)(t). Hence, by the positivity of Vα,β(t) (t ≥ 0) one can see

Combining with (10), (12) and the positivity of Uα,β(t) (t ≥ 0), it is easy to see Q u(1) ≤ Q u(2). Therefore,: [v(0), w(0)] → [v(0), w(0)] is monotone increasing. Next, let

then we can obtain two sequences {v(i)} and {w(i)} in [v(0), w(0)]. By the monotonicity of the operator Q one can see

Step 2. v(i) and w(i) are equicontinuous in [ -r, a].

In fact, for each u ϵ [v(0), w(0)], by (10) we only consider it on [0, a]. Without loss of generality, let 0 ≤ t1 < t2 ≤ a. By (10) one can see

Next, we check if ‖Ji‖ tend to 0 as t2 - t1 → 0 (i = 1, 2, 3, 4), which are not dependent on u ϵ [v(0), w(0)]. It is easy to see that J1 → 0 as t2 - t1 → 0 by Lemma 2(i). By (9) and (11) we can obtain

If t1 = 0 and 0 < t2 ≤ a, then it easy to see that J3 = 0. For t1 > 0 and ϵ > 0 small enough, by (9), (11) and Lemma 2(iv) we get that

Finally, by (9) and (11) we have

Therefore, ǁQu(t2)−Qu(t1)ǁ tends to 0 independently of u ∈ [v(0), w(0)] as t2−t1 → 0, which implies that Q : [v(0), w(0)] → [v(0), w(0)] is equicontinuous.

Step 3. {v(i)(t)} and {w(i)(t)} are relatively compact on E for each t ∈ J.

Let Λ = {v(i)}, Π = {w(i)} and Λ0 = Λ ∪ {v(0)}, Π0 = Π ∪ {w(0)}. Obviously, Λ(t) = (QΛ0)(t) and Π(t) = (QΠ0)(t) for t ∈ J. In view of the fact that v(i)(t) = w(i)(t) = ϕ(t) for t ∈ [−r, 0], {v(i)(t)} and {w(i)(t)} are relatively compact on E for t ∈ [−r, 0].

Let 0 < t ≤ a be fixed. For any ε ∈ (0, t) and δ > 0, define a set Qε,δΛ0(t) by

Hence, from the compactness of U α,β(t) and Sα(εαδ) it follows that Qε,δΛ0(t) is relatively compact in E for each δ > 0 and ε ϵ (0, t). Moreover, for every v(i) ϵ Λ0 and 0 < t ≤ a, one can find

Thus, the set ( QΛ0)(t) is relatively compact, which implies that v(i)(t) is relatively compact on E for 0 < t ≤ a. Thus, v(i)(t) is relatively compact on E for t ϵ J. Similarly, one can obtain that w(i)(t) is relatively compact on E for t ϵ J.

As we all know, the above argument and the Arzela–Ascoli theorem guarantee that {v(i)} and {w(i)} are relatively compact in C(J, E). Hence, there are convergent subsequences in {v(i)} and {w(i)}, respectively. Combining the normality of the cone KC and the monotonicity, we obtain that {v(i)} and {w(i)} themselves are convergent, namely, there exist u, u ∈ C(J, E) such that limi→∞ v(i) = u and limi→∞ w(i) = u.

Hence, taking i → ∞ in (13), we have

Therefore, u, u ∈ Ω are fixed points of Q, which are mild solutions of Eq. (6).

Step 4. Maximal and minimal properties of u, u.

Let ũ ∈ [v0 w0] be a fixed point of Q, then v(0)(t) ≤ u˜(t) ≤ w(0)(t) for each t ∈ J, and

Taking i ∞ in (14), we can obtain u ≤ ũ ≤ ū, which implies that ū, u are maximal and minimal mild solutions of Eq. (6).

In the above works, the key assumption (H1) (the monotone on the history function of the nonlinear function) is employed. However, we hope that the nonlinear function is quasimonotonicity. In this case, the results have more extensive application background.

In fact, we find that if Eq. (6) has the upper and lower solutions w(0), v(0) with v(0) ≤ w(0) and

(H2) for any u(1), u(2) [v(0), w(0)] with u(2) ≥ u(1), there exists a sufficiently small constant C0 > 0 such that

then condition (H1) can be replaced by

(H3) there are nonnegative constants C1, C2 such that

for any t ∈ [0, a], x1, x2 ∈ E and φ1, φ2 ∈ B with v(0)(t) ≤ x1 ≤ x2 ≤

w(0)(t) and vt(0) ≤ φ1 ≤ φ2 ≤ wt (0).

In fact, for every t ∈ [0, a] and u(1), u(2) ∈ [v(0), w(0)] satisfying u(1) ≤ u(2), one can obtain that v(0)(t) ≤ u(1)(t) ≤ u(2)(t) ≤ w(0)(t), vt(0) ≤ ut(1) ≤ ut(2) ≤ wt(0). From conditions (H2) and (H3) it follows that

Hence, we can obtain the following result from Theorem 1.

Theorem 2. Assume that Eq. (6) has upper and lower solutions w(0), v(0) with v(0)≤ w(0). If F : [0, a] × E × B → E is continuous and satisfies (H2) and (H3), then Eq. (6) has maximal and minimal mild solutions ū, u ∈ [v(0), w(0)].

Remark. Obviously, condition (H2) is easy to satisfy, and condition (H3) weakens condition (H1). Thus, Theorem 2 partially improve Theorem 1.

In the end of this section, we study the uniqueness of the mild solution for Eq. (6).

Theorem 3. Assume that Eq. (6) has upper and lower solutions w(0), v(0) with v(0)≤ w(0). If F : [0, a] × E × B → E is continuous and satisfies (H2), (H3) and

(H4) there exist positive constants L1, L2 such that, for any x1, x2 ∈ E and φ1, φ2 ∈B with v(0)(t) ≤ x1 ≤ x2 ≤ w(0)(t), vt(0) ≤ φ1 ≤ φ2 ≤ wt(0),

then Eq. (6) has a unique mild solution in [v(0), w(0)].

Proof. From Theorem 2 we can assert that Eq. (6) has maximal and minimal mild solutions ū, u ∈ [v(0), w(0)], which are fixed points of Q defined by (10).

By (10) one can find that ū − u ≡ θ for t ∈ [−r, 0]. On the other hand, for t ∈ [0, a],

where C = C1 + C2/C0. Hence, from the normality of the cone K it follows that for any t ∈ [0, a],

From Lemma 3 it follows that u(t) = ū(t) for t [0, a]. Therefore, u = ū is the unique mild solution of Eq. (6) in [v(0), w(0)].

4 Results for the time-space fractional diffusion equation

In the following, we will apply our abstract results to prove the existence and uniqueness of the mild solutions for the time-space fractional delayed diffusion equation with fractional Laplacian (2).

Theorem 4. Let f (x, t, 0, 0) ≥ 0 for any (x, t) ∈ Ω × [0, a], and let w = w(x, t) ∈ C(Ω × [−r, a]) ∩ C2,α(Ω × [0, a]) be a nonnegative function satisfying

If for any x ∈ Ω, t ∈ [0, a], τ ∈ [−r, 0], and 0 ≤ u1(x, t) ≤ u2(x, t) ≤ w(x, t), (A1) there is a constant c0 > 0 such that

(A2) there are constant c1, c2 > 0 such that

then semilinear time-space fractional diffusion equation boundary value problem with delay (2) has maximal and

minimal mild solutions ū, u C([ - r, a], L2(Ω)) between 0 and w.

Proof. Let E = L2(Ω) with the L2-norm , K = u E u(x) ≥ 0, a.e. x Ω , which defines a partial ordering “≤” on E. Thus, E is an ordered Banach space, and the positive cone K is a normal regeneration cone.

Define operator A : D(A) ⊂ E → E as follows:

From [2] it follows that A is a selfadjoint operator, which generates a uniformly bounded analytic semigroup T (t) (t ≥ 0) in E. Specially, T (t) (t ≥ 0) is contractive in E, hence, ‖T (t)‖ ≤ 1 for every t ≥ 0. Furthermore, we assume that λ1 is the first eigenvalue of operator A, then λ1 > 0 from [1, Thm. 1.16]. On the other hand, λI + A has a positive bounded inverse operator (λI + A)−1 for λ > 0, it follows that T (t) (t ≥ 0) is a positive C0-semigroup. Since the operator A has compact resolvent in L2(Ω), thus T (t) (t ≥ 0) is a compact semigroup (see [25]).

Obviously, the fractional power Aβ of the nonnegative operator A is well defined by (3). Therefore, based on the argument in Section 2 and the properties of T (t) (t ≥ 0) generated by A, one can deduce that Aβ generates a positive, compact, uniformly bounded and analytic semigroup Tβ(t) (t ≥ 0) on E, and ‖Tβ(t)‖ ≤ 1 for all t ≥ 0.

Set u(t)(ξ) = u(ξ, t), u(t + τ )(ξ) = u(ξ, t + τ ), and

thus, the boundary value problem (2) can be rewritten as follows:

From the assumptions of the function f we can deduce that the function F : [0, a] x E x E → E defined by (15) is continuous and satisfies condition (H1). And from the assumptions one can find that v0 ≡ 0, and w0 = w(ξ, t) ≥ 0 are lower and upper solutions of problem (16), respectively. By conditions (A1) and (A2) we can deduce that conditions (H2) and (H3) hold. Therefore, by Theorem 2 one can find that Eq. (16) has minimal and maximal mild solutions u, ū ∈ C([−r, a], E).

According to the proof of the above theorem, it is not difficult to obtain the following uniqueness results.

Theorem 5. Under the assumptions of Theorem 4, if the following condition holds: (A3) there are positive constants c1, c2 such that

then the semilinear time-space fractional diffusion equation boundary value problem with delay (2) has a unique mild solution u∗ ∈ C([−r, a], E) between 0 and w.

References

1. H. Amann, Nonlinear operators in ordered banach spaces and some applications to nonlinear boundary value problems, in Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Springer, Berlin, 1976, pp. 1–55, https:

2. H. Amann, Periodic solutions of semilinear parabolic equations, in Nonlinear Analysis: Col- lection of Papers in Honor of Erich H. Rothe, Academic Press, New York, 1978, pp. 1–29, https://doi.org/10.1016/B978-0-12-165550-1.50007-0.

3. B. Baeumer, M. Meerschaert, E. Nane, Space-time duality for fractional diffusion, J. Appl. Probab., 46:1100–1115, 2009, https://doi.org/10.1017/s0021900200006161.

4. A. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pac. J. Math., 10:419–437, 1960, https://doi.org/10.2140/pjm.1960.10.419.

5. D. Baleanu, M. Inc, A. Yusuf, A. Aliyu, Lie symmetry analysis, exact solutions and conser- vation laws for the time fractional modified Zakharov–Kuznetsov equation, Nonlinear Anal. Model. Control, 22(6):861–876, 2017, https://doi.org/10.15388/NA.2017.6.9.

6. A. Bonito, J. Borthagaray, R. Nochetto, E. Otárola, A.J. Salgado, Numerical methods for fractional diffusion, Comput. Vis. Sci., 19:19–46, 2018, https://doi.org/10.1007/ s00791-018-0289-y.

7. A. Bonito, W. Lei, J. Pasciak, The approximation of parabolic equations involving fractional powers of elliptic operators, J. Comput. Appl. Math., 315:32–48, 2017, https://doi.org/10.1016/j.cam.2016.10.016.

8. P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 711-728:391–404, 2014, https:

9. Z. Chen, M. Meerschaert, E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., .:479–488, 2012, https://doi.org/10.1016/j.jmaa.2012. 04.032.

10. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010,

11. A. Hanyga, Multi-dimensional solutions of space-time-fractional diffusion equations, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci., 458:429–450, 2002, https://doi.org/10.2307/3067353.

12. J. Jia, K. Li, Maximum principles for a time-space fractional diffusion equation, Appl. Math. Lett., 62:23–28, 2016, https://doi.org/10.1016/j.aml.2016.06.010.

13. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, https://doi.org/10.1016/S0304- 0208(06)80001-0.

14. M. Kwas´nicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20:7–51, 2017, https://doi.org/10.1515/fca-2017-0002.

15. Q. Li, Y. Li, Monotone iterative technique for second order delayed periodic problem in Banach spaces, Appl. Math. Comput., 270:654–664, 2015, https://doi.org/10.1016/j.amc.2015.08.070.

16. Q. Li, Y. Li, P. Chen, Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay, Kodai Math. J., 39:276–289, 2016, https://doi.org/10.2996/ kmj/1467830137.

17 M.M. Meerschaert, M. Ainsworth, G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404:109009, 2020, https:

18. Y. Luchko, On some new properties of the fundamental solution to the multi-dimensional space-and time-fractional diffusion-wave equation, Mathematics, .(4):76, 2017, https:

19. Y. Luchko, Subordination principles for themulti-dimensional space-time-fractional diffusion wave equation, Theory Probab. Math. Stat., 98:121–141, 2018, https://doi.org/10.1090/tpms/1067.

20. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., .:153–192, 2001.

21. C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Stud., Vol. 187, North-Holland, Amsterdam, 2001.

K. Miller, B. Ross, An Introduction To the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993, https://doi.org/10.1186/1471-2164-13- 616 .

23. R. Nagel, One-Parameter Semigroups of Positive Operators, Springer, Berlin, 1986, https:

24. J. Padgett, The quenching of solutions to time-space fractional Kawarada problems, Comput. Math. Appl., 76:1583–1592, 2018, https://doi.org/10.1016/j.camwa.2018.07.009.

25. A. Pazy, Semigroup of linear operators and applications to partial differential equations, Springer, New York, 1993, https://doi.org/10.1007/978-1-4612-5561-1.

26. B. Rubin, Fractional Integrals and Potentials, Pitman Monogr. Surv. Pure Appl. Math., Vol. 82, Longman, Harlow, 1996.

27. U. Skwara, L. Mateus, R. Filipe, Superdiffusion and epidemiological spreading, Ecol. Complexity, 36:168–183, 2018, https://doi.org/10.1016/j.ecocom.2018.07.006.

28. H. Sun, Y. Zhang, D. Baleanu, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64:213–231, 2018, https://doi.org/10.1016/j.cnsns.2018.04.019.

29. L. Toniazzi, Stochastic classical solutions for space-time fractional evolution equations on a bounded domain, J. Math. Anal. Appl., 469:594–622, 2019, https://doi.org/10.1016/j.jmaa.2018.09.030.

30. J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal., Real World Appl., 12:263–272, 2011, https://doi.org/10.1016/j.nonrwa.2010.06.013.

31. H. Ye, J. Gao, Y. Ding, A generalized gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328:1075–1081, 2007, https://doi.org/10.1016/j.jmaa.2006.05.061.

32. K. Yosida, Functional Analysis, Springer, Berlin, 1965, https://doi.org/10.1007/ 978-3-642-61859-8.

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