On the sub–di usion fractional initial value problem with time variable order

As early as 1972, G. Scarpi [21], based on an earlier work of Smit and de Vries [22] concerning rheological models with fractional derivatives, proposed an evolutive rheological model with fractional derivative with order varying in time as an ultime generalization. His de nition seems to us appealing; we adopt it here. More than forty years later, in the contributions to Round Table Discussion "Fractional Calculus: Quo Vadimus? (Where arewegoing?) held at ICFDA2014Catania (Italy), 23-25 June 2014,M. Fabrizio commentedon fractional derivatives of variable order in the following words: I believe that a promising eld of application of the fractional calculus is to study problems where the α-coe cient (order) of the fractional derivative is varying with time. In spite of several theoretical and numerical papers on this topic, it seems to me that applications are not yet all investigated. In fact, if in continuum mechanics we consider a visco-elastic materials described by a fractional derivative, its α-order is constant and assigns the constitutive law of viscoelasticity. In many problems we observe a change in the nature of the material due to the deformation or simply to the time. Such variation leads to a change of the α-exponent, that can especially be considered as a new variable of the problem. This can be an important step and a qualitative leap for the applications that the fractional derivatives can help to resolve. Phenomenawith α-order variable are evident in fatigue, in the plasticity but also for magnetic hysteresis in electromagnetism and in some problems of phase transitions. So I think that considering fractional derivatives of variable order may be a promising way to apply fractional calculus to the above complex phenomena. Let us mention that in the literature one may nd several de nitions of time variable order fractional derivative [4], [5], [23], [25], [12], [13], [14], and even the order of derivation varying in time and the unknown of the system [20] to cite but a few. However, it seems to us that the one that should be adopted is the de nition


Introduction
As early as 1972, G. Scarpi [21], based on an earlier work of Smit and de Vries [22] concerning rheological models with fractional derivatives, proposed an evolutive rheological model with fractional derivative with order varying in time as an ultime generalization. His de nition seems to us appealing; we adopt it here. More than forty years later, in the contributions to Round Table Discussion "Fractional Calculus: Quo Vadimus? (Where are we going?) held at ICFDA 2014 Catania (Italy), 23-25 June 2014, M. Fabrizio commented on fractional derivatives of variable order in the following words: I believe that a promising eld of application of the fractional calculus is to study problems where the α-coe cient (order) of the fractional derivative is varying with time. In spite of several theoretical and numerical papers on this topic, it seems to me that applications are not yet all investigated. In fact, if in continuum mechanics we consider a visco-elastic materials described by a fractional derivative, its α-order is constant and assigns the constitutive law of viscoelasticity. In many problems we observe a change in the nature of the material due to the deformation or simply to the time. Such variation leads to a change of the α-exponent, that can especially be considered as a new variable of the problem. This can be an important step and a qualitative leap for the applications that the fractional derivatives can help to resolve. Phenomena with α-order variable are evident in fatigue, in the plasticity but also for magnetic hysteresis in electromagnetism and in some problems of phase transitions. So I think that considering fractional derivatives of variable order may be a promising way to apply fractional calculus to the above complex phenomena. Let us mention that in the literature one may nd several de nitions of time variable order fractional derivative [4], [5], [23], [25], [12], [13], [14], and even the order of derivation varying in time and the unknown of the system [20] to cite but a few. However, it seems to us that the one that should be adopted is the de nition based on the Laplace transform. Here after recalling the de nition proposed by Scarpi, we address the important Leibniz rule from which we derive an important inequality that is useful to obtaining estimates for fractional di erential equations as it was demonstrated for the case of a constant fractional order in [1]. Before stating and proving our results, we recall some de nitions and preliminany results.

Background
Let us rst recall the de nition we consider along the paper; we will denote by L and L − the Laplace transform and the inverse Laplace transform operators, respectively. Let α : [ , +∞) → ( , ) be a function that admits the Laplace transform denoted byα(z) := L(α)(z), for z belonging to certain complex domain D containing C + a := {z ∈ C : Re(z) ≥ a} (assume a > ). For instance, a variety of piecewise continuous functions of exponential growth. Moreover, assume now and hereafter that t > means to say a.e. t > .
De nition 1. Given f ∈ L ([ , +∞)), we de ne where k : [ , +∞) → R is given in terms of the Laplace transform as where D ′ is a certain complex domain such that C + a ⊂ D ′ , Γ is a complex path according to the Bronwich formula for the inversion of the Laplace transform whose precise choice is discussed below, and k(t) vanishes for t < . Note that in the case of α(t) = α = constant, the de nition (2.1) matches the very well known de nition of fractional integral of order α in the sense of Riemann-Liouville, this is why this de nition is often known as the fractional integral of variable order in the sense of Riemann-Liouville. Now we are in a position to de ne the fractional derivative of variable order α(t) (FDVO). To this end there are typically two di erent ways, both in terms of the de nition of the fractional integral (2.1)- (2.2).
In spite of the regularity does not a matter here, to be precise the functions involved in the following results are required to belong to a convenient Sobolev space. In fact denote W m,p ([ , +∞)) := f ∈ L p ([ , +∞)) : ∂ n f ∈ L p ([ , +∞)), ≤ n ≤ m .

4)
where f ′ stands for the rst time derivative of f , and k(t) is given by  [10], among many others references). In order to keep the notation as simple as possible and while not confusing, we denote again the Laplace transform of the convolution kernel by K(z). 2nd.-Alternatively one may de ne the FDVO, for < α(t) < for f di erentiable, as where k is de ned by (2.5).
Several comments related to the previous proposed de nition for the FDVO can be made. First of all note that, unlike what happens with the de nition (2.3), with the de nition (2.6) if f (t) = constant, then the fractional derivative turns out to be zero. Moreover, if f ( ) = , then both de nitions coincide. Although both de nitions show some di erences, the treatment turns out to be quite similar for both, as we will show below.
Moreover, based on a fact that has already been observed in [6,17,18], we can assert that (2.2) and (2.5) are meaningful under very weak restriction onα. In particular, let K(z) be a complex-valued or operator-valued function, analytic outside of a complex sector of angle θ, θ < π/ , denoted by and such that there exist M > and β ∈ R, satisfying As explained in [6,17,18], K(z) stands for the Laplace transform of a distribution k(t) in the real line, so that k(t) = , for t < , whose singular support is empty, or merely concentrated at t = (e.g. if β < ), and which is analytic for t > . In that case the inverse (distributional) Laplace transform, i.e. k(t), admits the integral representation for a convenient path Γ in the complex plane running outside the sector of analyticity S θ , e.g. one running parallel to the boundary of S θ , and with increasing imaginary part. We will precisely state below one of these paths, according to our interest for the proofs. In this context, let us recall the following lemma which has already been proven in [6], Lemma 2.1; it provides useful bounds for k(t).

Lemma 1. Let K(z) be a analytic function outside a complex sector S
Then there exists C > depending solely on M, β and θ (but not on t) such that the inverse Laplace transform of K(z), k(t), is bounded as follows Observe that bound (2.9) suggests the lack of regularity of k(t) at t = , if β < . Moreover, note that k(t) is locally integrable is β > . A more general case might be considered if instead of the sector S θ of analyticity, one considers a shifted sector a + S θ = {a + z : z ∈ S θ }, for a ≥ . However, since no noticeable additional di culties arise in that case, and for the sake of the simplicity of the presentation of results, we merely consider the case a = , i.e. S θ .
Before going ahead in the paper, we make some assumptions which will be required when studying the solutions of the sub-di usion initial value problems of variable order α(t) in Section 4. These assumptions are related to sectoriality angle θ, which will also a ect the linear operators considered there as it will be discussed in that section, and the fractional order α(t), and more particularly its Laplace transformα(z). In fact, assume the following: There exist R > large enough, and < m , m , C < , satisfying [H2]. The analyticity angle θ satis es Note that, by hypothesis [H1], we have that − m /( − C ) ≤ , so this hypothesis is meaningful.
On the other hand note that, under Hypotheses [H0]-[H2] we have for (2.5), Therefore these inequalities suggest a lack of regularity of k(t) as t → + , and that the function k(t) is locally integrable in R + . The same can be said for (2.2) related to the regularity and integrability since

Leibniz rule and integration-by-parts formula for fractional derivatives with time variable order
A Leibniz rule for fractional derivatives with constant order has already been provided in [1] as a generalization of the classical product rule for integer derivatives. It can be observed that this derivation rule (and others of fractional type one can nd in the literature) involve additional terms inspired by the non local character of the fractional derivatives, particularly in the case of FDVO. One of our main contributions in this paper is to go further in this generalization by extending such a property to the fractional derivatives, now with a variable order α(t).
On the one hand it is expected that the Leibniz rule we achieve will adopt di erent forms depending on the de nition of FDVO one considers. On the other hand the de nitions of FDVO we consider here may provide the simplest product rule among the ones provided by other de nitions. An expected and desirable fact is that the Leibniz rule one achieves for fractional derivatives and particularly the ones achieved here for FDVO should be coherent with the classical product rule. That is the case for the equalities derived in this section.
The next theorem stands for the main result of this section.
Proof. Consider rst the De nition (2.3). On the one hand, on the other hand Therefore, Moreover by (2.4) we have and the proof of the equality (2.3) ends. For the De nition (2.6), we merely recall that from which (3.2) straightforwardly follows, and the proof of the theorem ends.
Here we have to highlight several facts.  .2)) is satis ed in the particular case of α(t) ≡ α, < α < . It can be easily illustrated by taking for example f (t) = t p , g(t) = t q , p, q ∈ Z + , and any < α < . In particular, in the constant case α(t) ≡ α, < α < , the product rule achieved here perfectly matches with the one derived in Eq. 3.12, [2], (∂ α stands for the fractional derivative in the sense of Riemann-Liouville of order α > ). -Another relevant fact is that both equalities (3.1) and (3.2)) are coherent with the classical product rule, namely the classical product rule turns out to be the limit case as α(t) tends to the constant function α(t) ≡ . In that case the corresponding limit of the convolution kernel in the sense of distributions, and always according to de nition (2.5), turns out to be k(t) = δ (t) where δ (t) stands for Dirac's delta distribution with density concentrated at t = . In that manner, since k(t) = , for t < , then k(t)f (t)g(t) = , for t < . Moreover, according to the de nition (2.3) and in the limit Therefore it comes ∂ α(t) (h(t, ·))(t) = . Similarly, according to the de nition (2.6) and since in the limit both equalities (3.1) and (3.2) boil into the usual Leibniz rule (f (t)g(t)) ′ = f (t)g ′ (t) + f ′ (t)g(t), t > , as conjectured above.
-Finally, in spite of the inequality has been proven in [2] from the equality (3.3) for the case of fractional derivatives of constant order α > , in the case of the derivatives with time variable order considered here, the inequality is not satis ed in general. Only if we can guarantee that ∂ α(t) (h(t, ·))(t) − k(t)f (t)g(t) ≤ , (or simply ∂ α(t) (h(t, ·))(t) = depending on the de nition one adopts) for t > , the inequality (3.4) could become valid, although this is not obvious at all for a so general class of functions α(t) we are here considering, i.e., α(t) under the hypotheses made in Section 2.
From Theorem 1 several integration-by-parts formulas for FDVO can be straightforwardly derived. We show two of such formulas in the following corollaries. (3.5) and according to the De nition (2.6)

Corollary 2. Let f , g be two functions belonging to W , ([ , +∞)). According to the De nition (2.3) we have the following integration-by-parts formula for FDVO
where

k(t) is in both cases the kernel de ned in (2.5).
Proof. Let φ be a function belonging to L ([ , +∞)).
First of all, we show an equality that will be useful for the proof. In fact if we consider the De nition (2.3), then applying the Laplace transform we straightforwardly have that Note that this equality means that ∂ −α(t) is the inverse operator of ∂ α(t) . Moreover if α(t) ≡ and φ( ) = , then this equality coincides with the classical one.
In the same manner if one takes the de nition (2.6) there is a subtle di erence, in fact applying again the Laplace transform we have This formula perfectly matches with the classical one. Now consider De nition (2.3). Thanks to Theorem 1 and equality (3.7), we have and the equality (3.5) follows (recall that h(t, t) = , for t ≥ ).
In the same fashion (3.6) is obtained.
Notice that, by the same arguments as in Theorem 1, as α(t) tends to the constant function α(t) ≡ , the kernel k(t) tends in the distributional sense to Dirac's delta distribution δ (t), and therefore the term ∂ −α(t) k(t)f (t)g(t) tends to δ (t)f ( )g( ) = therefore the equality stated in Corollary 2 reads Therefore, if f ( )g( ) = (in particular if f ( ) = g( ) = ), then (3.9) is coherent with the classical integrationby-parts formula. However according to the De nition (2.6) although f ( ) = g( ) = , the formulation is not coherent with classical case since h(t, ) ≠ .
An alternative form of the integration-by-parts formula for FDVO in given in the following corollary. This time the corollary is stated only for the De nition (2.3) since we did not nd the related formula for the De nition (2.6) coherent with the classical formula i.e. for α(t) ≡ .

Corollary 3. Let f , g be two functions belonging to W , ([ , +∞)). According to the De nition (2.3) we have the following integration-by-parts formula for FDVO
for t > , where k(t) is the kernel de ned in (2.5).
Proof. By Theorem 1, one has If one considers the De nition (2.3), then the equality (3.10) straightforwardly follows.
The equality (3.10) turns out to be coherent with the classical integration-by-parts formula as α(t) tends to the function since as above k(t) tends in the distributional sense to Dirac's delta distribution δ (t) and therefore, and T k(T − t)h(t, T) dt tends to h(T, T) = .
In that case, that is if α(t) tends to the constant function α(t) ≡ , then we have again the classical integrationby-parts formula

On the initial data for fractional ODE's . Formulation
In this section, we formulate precisely the sub-di usion initial value problem with time dependent order fractional derivatives with particular attention to the choice of the initial data. In fact, we here discuss how the initial data of the value problem has to be chosen in coherence with the regularity of the solution and the variable order of the fractional derivatives in the equation.
Another issue is the time regularity of the solution of the fractional initial value problems with non constant order. In the case of constant order α(t) = α the regularity has been already studied [6] in the context of the numerical solutions to such a kind of problems. Several properties of the fractional di usion equation with time dependent order, including the time regularity, have been already studied in [8] if < α(t) < . Here, we show the precise form of the non regular part of the solution as t → + in the case of < α(t) < . This is a crucial subject for instance when discretizing in time such a kind of equations since this lack of regularity restricts the order of convergence of the numerical schemes.
We conclude this section with the study of the asymptotic behavior of the solution of such a initial value problems.
For the shortness of the presentation and since no relevant di erences arise on the proofs, in this section we solely focus on the FDVO given by (2.3). Therefore consider the abstract time varying order fractional di erential equation where U(z) represents the Laplace transform of u(t), and K(z) is de ned by (2.5). Therefore, the Laplace transform in both(4.2) leads to In the same manner the Laplace transform in both sides of (4.1) leads to which will be consistent with the regularity of u(t) stated in Theorem 4 below, and advances the lack of regularity of u(t) at t = . Now, we are in a position to formulate precisely the fractional sub-di usion initial value problem with non constant order we study in this section. Here, we opt for a very general/abstract framework which is the one of the linear sectorial operators. In fact, consider where f ∈ L ([ , +∞), X), X stands for a complex Banach space, and the linear operator A : D(A) ⊂ X → X is a θ-sectorial operator, for < θ < π/ in X. Let us recall that a linear and closed operator A is θ-sectorial if its resolvent R(z) = (zI − A) − (where I is the identity operator in X) is analytic outside the sector S θ in the sense of (2.8) with β = , i.e. there exists M > , such that R(z) is analytic outside the sector S θ := {z ∈ C : | arg(−z)| < θ}, and satis es For the sake of the simplicity of the presentation, and without loss of generality, as we have discussed in Section 2 we consider a non shifted sector S θ (according to the notation in Section 2 a = ). Moreover if not confusion, the we denote by simplicity · instead of · X→X . Note that a lot of linear operators t in that general framework, e.g. complex scalars, nite dimensional operators like matrices (e.g. the ones coming out from most of the spatial discretization of elliptic operators), or in nite dimensional operators (like Laplacian ∆, or fractional powers of the Laplacian −(−∆) β with β > ), and so on.
Moreover, the assumptions [H0]-[H2] arising from the de nition of FDVO we have opted for, force us to make additional assumptions now for A. In fact assume [H3]. A is a θ-sectorial operator such that the sectoriality angle θ obeys the Hypothesis [H2].

. Regularity
This section is devoted to study the regularity of the solution of (4.6) under the hypotheses [H0]-[H3] stated in previous sections, in particular the statement of the theorem presented in this section shows how far goes the lack of regularity at t = of the mild solution of (4.6) or in other words how is the structure of the singularity of the solution as t → + .
Under the Hypotheses [H0]-[H3], the resolvent (zI − A) − exists and the equality (4.4) is now meaningful, for z ∉ S θ . But even more, the same hypotheses unable us to apply the Bromwich formula for the inverse Laplace transform to obtain a closed form of the mild solution of (4.6) u(t) = πi Γ e zt (zK(z) − A) − (u +f (z)) dz = πi Γ e zt (z zα(z) − A) − (u +f (z)) dz, t > , (4.7) where Γ is a suitable complex path connecting −i∞ and +i∞ with increasing imaginary part. We should not get confused with the complex path Γ considered in (2.3)-(2.5) and the one in (4.7) in spite of the underlaying ideas to de ne one of these paths are the same in both cases. Let us de ne for (4.7) one of these complex paths that is suitable for us, in particular according to Hypotheses It is straightforward to see that the path Γ de ned by (4.8)-(4.9) satis es the following properties: zK(z) ∉ S θ for z ∈ Γ, Γ keeps outside of S θ , and the real part of z is negative for all z ∈ Γ . The representation (4.7) is now completely determined once determined the complex path (4.8)-(4.9), and the mild solution of (4.6) can be written in terms of the corresponding evolution operator, i.e. we can write where E(t) := πi Γ e tz (z zα(z) − A) − dz, t > , (4.10) where Γ is de ned by (4.8)-(4.9). Note that it serves as a proof of the well posedness of (4.6). Next theorem shows the regularity of the solution of (4.6) depending on the regularity of the initial data u , and source term f (t).
whatever the initial data u is merely belonging to X, i.e. without any regularity requirement.