On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on RN

Abstract:We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on RN , which is derived from a non-Newtonain uid for a generalized second grade uid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very e ective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on theGagliardo-Nirenberg inequalities, operator theory, standard xed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.


Introduction
Fractional calculus has proved a powerful tool to describe the viscoelasticity of uids and anomalous di usion phenomena, such as the constitutive relationship of the uid models [21], basic random walk models [25], and so on. Besides in recent years, mainly due to the nonlocal characteristic of the fractional derivative, there are some excellent works on stochastic processes driven by fractional Brownian motion [13] and on physical phenomena like inverse problems for heat equation [19] and memory e ect [2]. It is worth mentioning some solid works about time-fractional derivatives [1,4,7,12,14,17,[37][38][39] and space-fractional derivatives [15,16,33] and the references therein, most of conclusions in these works commuted with fractional models are quite di erent from the situation of integer derivative, for instance, decay and asymptotical behaviors, blow-up analysis, well-posedness analysis, stability and Liouville property etc..
It is known that many signi cant complex media are non-Newtonian and exhibit time-dependent behavior of thixotropy and rheopecty [23,28]. The behavior in non-Newtonian uid often follows the power law [24]. Time-dependent non-Newtonian properties are more closely linked to fractional viscoelasticity than previously thought. Especially in a generalized second grade uid, it is also more di cult to construct a simple mathematical model for describing many di erent behavior of non-Newtonian uids. Form this physics point of view, in the second grade uid, the employed constitutive relationship has the following form: where σ is the Cauchy stress tensor, p is the hydrostatic pressure, I is the identity tensor. µ ≥ , ϱ and ϱ are normal stress moduli. ϵ and ϵ are the kinematical tensors de ned by where V is the velocity, ∇ is the gradient operator and the superscript T denotes a transpose operation. When we consider the time-dependent of time derivative in the kinematical tensor ϵ , generally the constitutive relationship of viscoelastic second grade uids has the form as follows: where ∂ α t is the Riemann-Liouville fractional derivative of order α ∈ ( , ) de ned by provided the right-hand side is pointwise de ned, where Γ(·) stands for the Euler's Gamma function. The form of the model was selected for its ability to portray accurately the temperature distribution in a generalized second grade uid that subject to a linear ow on a heated at plate and within a heated edge, see [32]. In this paper, we concern with the following semilinear time-fractional Rayleigh-Stokes problem where ∆ is the Laplacain operator, f is a semilinear function and φ is a given initial condition in L p (R N ).
Such type of problems play a central role in describing the viscoelasticity of non-Newtonian uids behavior and characteristic. Many researchers showed a strong interest in this issue and they also obtained some satisfactory results. Here is a short description on the closely related works and comparision to our results. The Rayleigh-Stokes problem for a generalized second grade uid subject to a ow on a heated at plate and within a heated edge was introduced by Shen et al. [32] where their considered the exact solutions of the velocity and temperature elds. As for a viscous Newtonian uid, their revealed that the solutions of the Stokes' rst problem appear in the limiting for these exact solutions. The results in [32] were generalized later by Xue and Nie [34] in a porous half-space with a heated at plate, they also obtained an exact solution of the velocity eld and temperature elds, from which some classical results can be recovered. Both methods of two above papers are based on the Fourier sine transform and the fractional Laplace transform. For the smooth and nonsmooth initial data on a bounded domain Ω ⊂ R N (N = , , ), Bazhlekova et al. [5] considered the solutions of the homogeneous problem on C([ , T]; L (Ω)) ∩ C(( , T]; H (Ω) ∩ H (Ω)) for the initial value φ ∈ L (Ω), moreover, some operator theory and spectrum technique are used to also establish the related Sobolev regularity of the solutions. In the meantime, Bazhlekova [6] obtained a well-posedness result under the abstract analysis framework of a subordination identity relating to the solution operator associated with a bounded C -semigroup and a two parameters probability density function. Zhou and Wang [40] also established the existence results on C([ , T]; L (Ω))of nonlinear problem in operator theory on a bounded domain Ω with smooth boundary. As for some numerical solution of Rayleigh-Stokes problem with fractional derivatives, several scholars have considered and developed it, for example Zaky [36], Chen et al. [10,11], Yang and Jiang [35] etc.. Additionally, most of uid ows and transport processes use more distribution parameters to establish equations, the inverse problem of parameter identi cation has been proposed to deal with this matter, see Nguyen et al. [26,27].
In view of the results in above works, it is natural to consider whether the well-posedness result on R N can be extended to the mixed norm L p (L q ) spaces and whether it is possible to obtain the integrability of solutions. Unfortunately, it turns out that these extensions cannot be established by applying the technique of subordination principle in [6]. Moreover, some useful L p − L q inequality estimates about solution operator generated by the Eq. (1.1) are not easy to obtain, in which it is not immediately suitable to build an integral equation when we consider that the time-fractional derivative depends on all the past states, and also we cannot apply the approach of classical solution operators to derive the relevant estimates, while it is readily to achieve at the classical solution operators of type heat operator, like fractional di usion equations [22] and fractional Navier-Stokes equations [30]. To overcome the di culty from the e ect of time-fractional derivative, we propose a di erent technique to estimate the solution operator by means of the Gagliardo-Nirenberg inequality and generalized Gagliardo-Nirenberg inequality. For the proof, by using an admissible triplet concept that depends on the time-fractional derivative order α ∈ ( , ) and the exponents p, q in L p (L q ) spaces and their dimensions, we shall use the standard xed point argument to establish main well-posedness results. We also consider a contain special space to make the local existence, that is due to the decay exponent just depends on the order of time-fractional derivative deriving from the solution operator. We nd that the local solution will blow up in L r (R N ), and then the rate of the blow up solution may depend on the exponent of nonlinearity and r ≥ in L r (R N ). Also, based on the standard harmonic analysis methods, such as Marcinkiewicz interpolation theorem and doubly weighted Hardy-Littlewood-Sobolev inequality, some new conclusions likely the integrability of global mild solutions in Lebesgue spaces L µ ( , ∞; L r (R N )) are investigated.
This paper is organized as follows. In Section 2, we give some concepts about fractional calculus in Banach space and we introduce several useful analytic properties of Laplacian operator. By a rigorous analysis of solution operator S(t), we establish two crucial estimates that will be used throughout this paper. After introducing a de nition of mild solution in Section 3, the rst subsection shows the global and local wellposedness of the semilinear problem (1.1). Further, we obtain continuation and blow-up alternative of local mild solution of problem. In the last subsection, we show several integrability results of the global mild solution in Lebesgue space.

Preliminaries
Let (X, · ) be a Banach space and let L(X, Y) stand for the space of all linear bounded operators maps Banach space X into Banach space Y, we remark that C b (R+, X) stands for the space of bounded continuous functions which is de ned on R+ and takes values in X, equipped with the norm sup t∈R+ · X and C(J, X) stands for the space of continuous functions which is de ned on an interval J ⊆ R+ and takes values in X. If A is a linear closed operator, the symbols ρ(A) and σ(A) are called the resolvent set and the spectral set of A, respectively, identity R(λ; A) = (λI − A) − is the resolvent operator of A. We will denote by D(A α ), α ∈ ( , ), the fractional power spaces associated with the linear closed operator A.
An operator A is called the sectorial operator, if it follows the next concept.

De nition 2.1. Let A be a densely de ned linear closed operator on Banach space X, then A is called a sectorial
operator if there exist C > and θ ∈ ( , π/ ) such that Additionally, from [8, Theorem 2.3.2], it is not di cult to check that the Laplacian operator ∆ with maximal domain D(∆) = {u ∈ X : ∆u ∈ X} generates a bounded analytic semigroup of the spectral angle less than or equal to π/ on X := L p (R N ) with ≤ p < +∞. Moreover, the spectrum is given by σ(−∆) = [ , +∞) for < p < +∞. For δ > and θ ∈ ( , π/ ) we introduce the contour Γ δ,θ de ned by where the circular arc is oriented counterclockwise, and the two rays are oriented with an increasing imaginary part. In the sequel, let A = −∆, then A is a densely de ned linear closed operator on Banach space L p (R N ) with < p < +∞, we de ne a linear operator S(t) by means of Dunford integral as follows Remark 2.1. It is worth noting that the author [6] applied the technique of the subordination principle to study the solution operator in (2.1) of problem (1.1), that is,

where T(t) is a bounded C -semigroup and function ϕ(t, τ) is a probability density function with respect to both variables t and τ, that is
Nevertheless, we also nd that it is hard to get the

estimate of AS(t) on a Banach space unless this estimate AT(t) L(X) ≤ M may be valid for all t ≥ , constant M > under the bounded analytic semigroup of T(t). From this point of view, we can not apply the subordination principle to estimate the operator de ned as in (2.1) in the current paper.
Recall that for any θ > It should be noticed that estimates of the operator S(t) are standard in the theory of analytic semigroups as follows.
Lemma 2.1. [3, Lemma 4.1.1] Given θ ∈ ( , π/ ), let C be an arbitrary piecewise smooth simple curve in Σ θ+π/ running from ∞e −i(θ+π/ ) to ∞e i(θ+π/ ) , and let X be a Banach space. Suppose that the map f : Σ θ+π/ ×X×R + → X has the following properties: Then , and moreover there exists C > such that AS(t)x ∈ C(( , ∞); X), we have Proof. Let t > , θ ∈ ( , π/ ), δ > . We choose δ = /t, since operator −A generates a bounded analytic semigroup of the spectral angle is less than or equal to π/ , i.e., for any θ ∈ ( , π/ ) where Σ θ+π/ ⊂ ρ(−A). As a similarly approach in [5, Lemma 2.1], we conclude that g(z) ∈ Σ θ+π/ and |g(z)| ≤ M|z| −α for any z ∈ Σ θ+π/ , and thus from Lemma 2.2 we get S(t) L(X) ≤ M and S(t)x ∈ C(( , ∞); X) which shows the well-de ned part for t > . Additionally, for the Laplace transform of S(t), by virtue of Fubini's theorem and Cauchy's integral formula, we obtain for λ > , Consequently, in order to prove the limit point at t = in S(t)x for x ∈ X, the similar technique as in [31, Moreover, by using the identity Thus, the estimate (2.2) of AS(t) for t > is given by where θ = π/ − θ, Mα is a positive constant and it may depend on M, α and θ. Consequently, it follows that The proof is completed.
The inequality in (2.2) enables us to get another estimate about S(t)x in fractional power spaces. To do this, we need the following inequality.
Proof. This conclusion is an immediate result of Lemma 2.2 and Lemma 2.3. So, we omit it. Now, we introduce the concept of admissible triplet.

De nition 2.2.
We call (p, q, µ) as an admissible triplet with respect to α ∈ ( , ) if [20] where they concerned with space-time estimates to a fractional integro-di erential equation, in the meantime, one nds that this concept matches the L p − L q estimates of operator S(t) appropriately, see below lemma 2.5. Furthermore, it is worth noting that µ = µ(p, q) is completely determined by p and q.

Remark 2.2. For the de nition of admissible triplet, it is inspired by
Thenceforth, we give some useful L p − L q estimates about the linear operator S(t).
Lemma 2.5. The operator S(t) has the following properties: Proof. Using the classical Gagliardo-Nirenberg inequality, we know that there exists a constant C > such that taking the exponents of p, q into above inequality, we immediately obtain the L p − L q estimate of operator S(t). Hence, we have showed (i).
On the other hand, similarly, by virtue of the Gagliardo-Nirenberg inequality of fractional version, (see e.g. [18,Corollary 2.3.]), in view of Lemma 2.4, there exists a constant C > such that where q = θ p − N + ( − θ) p for any θ ∈ ( , ). Thus, the L p − L q estimate of A S(t)v follows. The proof of (ii) is completed.
In the sequel, we set a function W f with respect to f ∈ L ( , T; X) with any T > (or T = +∞), given by in which we will prove some properties of this function. Proof. Observe that for any t , t ∈ [ , T] with t < t , we have Since f ∈ L ( , T; X) and from Lemma 2.2, it follows that Additionally, we have which is integrable in L ( , t; X). By virtue of S(t)f (·) ∈ C([ , T]; X), we thus conclude that W f (·) ∈ C([ , T]; X) by Lebesgue's dominated convergence theorem. Next, for ω ∈ ( , ], we know ( − α)ω ∈ ( , ). By Lemma 2.4 we obtain which tends to zero as t → t . On the other hand, we have which is integrable in L ( , T; X). By virtue of A ω S(t)f (·) ∈ C(( , T]; X), we thus conclude that A ω W f (·) ∈ C([ , T]; X) by Lebesgue's dominated convergence theorem.
By Lemma 2.5, for t , t ∈ ( , T] with t < t , we see from which tends to zero as t → t by the properties of incomplete Beta function. Moreover, which is integrable in L ( , t ). Therefore, it follows from the similar method that W f (·) ∈ C(( , T]; L r (R N )).
In addition, if ξ < − N( −α) p − r , then it is easy to check that there exists a constant C > such that This implies that W f (t) tends to zero as t → in L r (R N ). Thus, W f (·) ∈ C([ , T]; L r (R N )). The proof is completed.

Well-posedness
Let u be a solution of problem (1.1), taking the Laplace transform into (1.1) yieldŝ by means of the inverse Laplace transform, we thus derive an integral representation of problem (1.1) by In the sequel, the well-posedness of problem (1.1) will be considered, in order to achieve this goal, the following general hypothesis of the semilinear function introduced by [9] will be also considered. Let r ′ , r be the conjugate indices.
(Hf)We suppose that f ( ) = and f : Additionally, we suppose that there exist constants σ ≥ and K > such that for all u, v ∈ L r (R N ). We rst consider the case T = +∞, i.e., the global well-posedness of the problem for mild solutions. For any α ∈ ( , ), let (p, r, µ) be an admissible triplet such that < r ′ ≤ p < r ≤ +∞, consider the Banach space  Proof. Let ε > and set Ωε = {u ∈ Xpr : u Xpr ≤ ε}.
It is easy to see that Ωε is a closed ball of Xpr with center 0 and radius ε. De ne the operator Φ in Ωε as The proof of the existence of unique global solution to problem (1.1) is based on the contraction mapping technique. From this point, we shall need some estimates which comes from this argument, we recall Lemma 2.2 and Lemma 2.5, it follows that there exists a constant C > such that Let us de ne λ := ε/( C) and observe that φ ∈ L p (R N ) with φ L p (R N ) ≤ λ, we thus get S(t)φ ∈ Xpr. Moreover, for any u ∈ Ωε, from the hypothesis (Hf) we deduce that Form the choice of µ that also determined by (p, r), we get which implies that N( −α) r ′ − p < and σ + < µ, combined with Lemma 2.5, we can easily derive the following estimates where B(·, ·) stands for Beta function, ϑ = (σ + )/µ ∈ ( , ), and − N( −α) r ′ − p = ϑ. Consequently, one derives On the other hand, the choice of µ implies that N( −α) r ′ − r < , as the same way in above arguments, we have It follows that Noting that for ϑ = (σ + )/µ > for choosing ε ≤ σ+ CKB(σ/µ, −ϑ) /σ , we thus get W f (u) Xpr ≤ ε for u ∈ Ωε. Hence, by the same choice of ε, it yields In addition, we now concern with continuity properties of (3.2). By virtue of Lemma 2.2 and similarly to Lemma 2.6, we know that Φ(u) ∈ C([ , ∞); L p (R N ))∩C(( , ∞); L r (R N )). Consequently, operator Φ maps Ωε into itself.
From the assumption of f and Lemma 2.5, for any u, v ∈ Ωε, we further obtain that with a similar argument, we get For the choice of ε, we have which shows that Φ is a strict contraction on Ωε. Thus Φ has a xed point u, and it is the unique solution of problem (1.1). Next, it just remains to prove the continuous dependence upon the initial data. Letũ be another mild solution of problem (1.1) associated with initial data ψ ∈ L p (R N ). We perform as in (3.3) to get and then the continuous dependence follows. The proof is completed.  In particular, from the restriction on µ in Theorem 3.1 and the admissible triplet ( , N + , µ), the dimension N in Corollary 3.1 is N = −α for some suitable α ∈ ( , ). Now, let us turn to the case T > and discuss the local well-posedness of the problem. Let α ∈ ( , ) and (p, r, µ) be an admissible triplet such that < p ≤ r < +∞ and p = r ′ < N, and Consider the Banach space Ypr[T] of continuous functions v : [ , T] → L p (R N ) under this admissible triplet by satisfying Proof. De ne the operator Φ in B(R) a closed ball of Ypr[T] with center 0 and radius R > as in (3.2) and we take C φ L p (R N ) = R. From (Hf), assumption µ > σ + and the proof in Theorem 3.1, it is easy to check that Φ(u) belongs to L p (R N ) for u ∈ B(R). Moreover, there exists a constant C > such that Moreover, the assumption µ > σ + derives that µ > for σ ≥ , that is for ϑ = (σ + )/µ ∈ ( , ), which implies that On the other hand, from Lemma 2.5 (ii) we have For this purpose, we rst claim that for < p ≤ r and φ ∈ L p (R N ), then To see this, note that by Lemma 2.5 (i) the maps t µ S(t) : L p (R N ) → L r (R N ), t ∈ ( , T], are uniformly bounded and converge strongly to zero on the dense subset L r (R N ) in L p (R N ). Hence, we get the desired argument. Similarly, Lemma 2.5 (ii) shows that the maps t −α (−∆) S(t) are uniformly bounded form L p (R N ) to itself and converge to zero strongly as t → . This means that and For the choice of T , we get Consequently, it follows that lim t→ t µ u(t) = in L r (R N ) and lim t→ t −α (−∆) u(t) = in L p (R N ). The proof is completed.
for the assumption (Hf) and admissible triplet (p, p, +∞), where Xpr reduces to L p (R N ). Obviously, the above right-hand side inequality of integral term is not integrable for all t ∈ [ , +∞). Thus, the case p = r is not valid on this argument. However, let p = r, it is easy to show that the local solution will belong to L ∞ ( , T; L (R N )) in Theorem 3.2 for some T > .
In the sequel, we establish the continuation and a blow-up alternative.
Proof. Since the assumptions of Theorem 3.
We note that Lemma 2.5 implies that the rst term of the right hand side of (3.5) is in Ypr [T], and then it goes to zero as t → T * . For this reason, we can choose Ta so close to T * such that S(·)φ − S(T * )φ Ypr[T] ≤ R/ . For the second term, we have as t → T * by the property of incomplete Bata function for ϑ = (σ + )/µ. Similarly, as t → T * . Therefore, as for the second term, we know that it belongs Ypr[T] and we can choose T b so close to T * such that its norm is less than R/ . As similar arguments, the last term of the right hand side of (3.5) belongs Ypr[T], moreover from Lemma 2.5 and Lebesgue's dominated convergence theorem can be applied to prove that its norm is less than R/ when we choose Tc so close to T * . Now, let T = min{Ta , In the same way, we can prove that Φ is a contraction on Suppose by contradiction that Tmax < ∞ and there exists a constant C > such that t µ u(t) L r (R N ) ≤ C and t −α (−∆) u(t) L p (R N ) ≤ C for all t ∈ ( , Tmax). Next, consider a sequence of positive real number {tn} ∞ n= satisfying tn → Tmax as n → ∞, we will verify that the sequence {u(tn)} ∞ n= belongs to L r (R N ). Let us show that the sequence {u(tn)} ∞ n= is a Cauchy sequence in L r (R N ). Indeed, for < t i < t j < Tmax, we get Therefore, the same reasoning used to estimate (3.5) gives that Hence, the limit lim n→∞ u(tn) := u(Tmax) exists in L r (R N ). Similarly, we can check the limit lim n→∞ (−∆) u(tn) in L p (R N ). Therefore, u(Tmax) and (−∆) u(Tmax) exist in Ypr [T]. As the before results in this theorem, the mild solution of problem (1.1) contradicts the maximality of Tmax. Thus, we may de ne the maximal mild solution u of problem (1.1) on the interval [ , Tmax). If we consider u(t ) as the initial value for some < t < Tmax, so u(t ) L r (R N ) < +∞ for p = r = , by above arguments we can prolong this solution u at least on [t , t ], it follows from (3.4) and the xed point argument, we have for some C > , C u(t ) L r (R N ) + CKMσ(t − t ) −ϑ R σ+ ≤ R, for some t < Tmax. Observe that if ≤ t < Tmax and C u(t ) L r (R N ) < R, then (Tmax − t ) −ϑ > R − C u(t ) L r (R N ) CKMσ R σ+ .
In fact, otherwise for some R > C u(t ) L r (R N ) and all t ∈ (t , Tmax) we would have which implies C u(t) L r (R N ) < R for all t ∈ (t , Tmax) by the previous arguments. However, this is impossible since u(t) L r (R N ) → +∞ as t → Tmax. Hence, choosing for example, R = C u(t ) L r (R N ) , we see that for < t < Tmax where C > is some new xed constant. Therefore, for the arbitrariness of t we obtain the desired blow-up rate estimate.

. Integrability of Solution
In this section, we will present the integrability of the global mild solution for current problem. For this purpose, we rst discuss the properties of the solution operator in L r ( , ∞; L q (R N )).
In the sequel, we divide the proof into two steps.
Together above arguments, we conclude that there is a unique global solution which belongs to L µ ( , ∞; L r (R N )). The proof of ii) is analogous, so we omit it. The proof is completed.
Con ict of interest statement: Authors state no con ict of interest.