Asymptotic profile for a two-terms time fractional diffusion problem

We consider the Cauchy-type problem associated to the time fractional partial differential equation: ∂tu+∂tβu-Δu=g(t,x),t>0,x∈Rnu(0,x)=u0(x),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+\partial _t^{\beta }u-\varDelta u=g(t,x), &{} t>0, \ x\in {\mathbb {R}}^n \\ u(0,x)=u_0(x), \end{array}\right. } \end{aligned}$$\end{document}with β∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (0,1)$$\end{document}, where the fractional derivative ∂tβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t^{\beta }$$\end{document} is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in Cb([0,∞),Hs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_b([0,\infty ),H^s)$$\end{document}. We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.


Introduction
In the present paper we consider the Cauchy-type problem for a fractional (in time) partial differential equation where ∂ β t u denotes the (forward) Caputo fractional derivative of order β ∈ (0, 1), with starting time 0, with respect to the time variable (see, for instance, [22]). Namely, for any t > 0 and x ∈ R n ; here, Γ (·) denotes the gamma function. In Theorem 1, we show that solutions to (1.1) are bounded in H s , with respect to t ∈ [0, ∞), namely, are t H s−2+ε and t β g ∈ L ∞ t H s−4 , where t = √ 1 + t 2 . In Theorem 2, in low space dimension n = 1, 2, 3, assuming initial data in L 1 ∩ H s with s ∈ [0, 2 − n/2) we prove that the asymptotic profile of the solution to (1.1) is independent of g, provided that suitable decay assumptions on g(t, ·) are satisfied.
As a corollary of this latter result, we investigate a class of nonlinear perturbations of the problem, for which global-in-time small data solutions exist and we show that their asymptotic profile is independent on the nonlinear perturbation.
One crucial property which allow us to get the previous results is a smoothing effect. In particular, the H s norm of u(t, ·) at any time t > 0 can be controlled by C(t) u 0 H s−2 , if g ≡ 0. However, C(t) → ∞ as t → 0 (see (3.10) and (3.12)). This effect is analogous to the smoothing effect of the heat equation and related parabolic equations, but it only allows to gain a finite amount of regularity. The smoothing effect also appears with respect to the inhomogeneous term g(t, ·), with a different singular power (see (3.9) and (3.13)), since the Duhamel's principle does not hold in classical sense, for Cauchy-type problems with Caputo fractional derivatives, as (1.1) (see later, Lemma 1).
The counterpart of this limited smoothing effect is a limited dissipative effect which appears at long time: higher order derivatives of the solution vanish as t → ∞ with a faster speed, but not faster than t −β . As for the smoothing effect, this limitation is due to the structure of the fundamental solution of the equation. In particular, to show the optimality of the decay estimates, at least in low space dimension, we describe the asymptotic profile of the solution, under suitable decay assumption on g(t, x). Under the moment condition M = 0, where (1.2) this profile is described by M K † 0 (t, x), where K † 0 is the fundamental solution to the homogeneous Cauchy-type problem for the sub-diffusive fractional equation (1.3) Explicitly, K † 0 (t, x) = t − nβ 2 K † 0 (1, t − β 2 x) is given by: |ξ | 2 |ξ | 4 + τ 2 + 2τ |ξ | 2 cos(βπ ) dτ ; (1.4) in particular, K † 0 (1, ·) belongs to H 2− n 2 (see for instance [10]). Here, and in the following, F denotes the Fourier transform operator acting on the space variable x, and f (t, ξ) = (F (t, ·))(ξ ). By representation (1.4), the limited amount of smoothing effect is motivated by the fact that |K † 0 (1, ξ)| ≈ |ξ | 2 ξ −4 . The dissipative-smoothing effect also appears in other evolution equations, for instance, in the case of strongly damped waves [41] (see also [8]), and of more general damped evolution equations [15]. However, those cases are more related to the heat equation and other diffusive equations, since the smoothing effect is not limited by ξ −2 . The study of the H s well-posedness for multi-point value problems for partial differential equations of fractional order similar to (1.1) is already faced, for instance, in [18].
The main difficulty in dealing with the equation in (1.1) is its lack of homogeneity. Theorems 1 and 2 are based on the representation formula provided by Lemma 1 for the solution to the Cauchy-type problem with λ > 0 and c 0 ∈ R.

Background
We refer to [22] or [33] for a deep study about the theory of fractional derivatives. It is well known that differential equations with fractional derivatives turned out to be suitable to describe in a very good way various physical phenomena in areas like rheology, biology, engineering, mathematical physics, etc. (see for instance [16,25,26,28,33] and the reference given therein). Open problem in this field is finding some easy and effective methods for solving such equations. Such problem becomes even more difficult when multiple fractional in time derivatives are involved in the equation.
In the literature some authors considered the two-term time fractional diffusion-wave equation of the type for b 1 , b 2 ∈ R, δ 1 , δ 2 > 0 and F ≡ 0 or F nonlinear; then, they investigate the existence of solution to the Cauchy-type problem associated to (1.6) in suitable spaces, under given assumptions on the exponents δ 1 and δ 2 and on the function F. A deep review can be found for instance in [43]; here, the authors find the upper viscosity solutions to (1.6) for b 1 + b 2 = 1, c = 1 and δ 1 , δ 2 ∈ (0, 2), considering a nonlinear lipschitz term F, in the L p (R n ) framework, for 1 ≤ p ≤ ∞. Equation (1.6) with δ 1 = 2δ 2 is known as the time-fractional telegraph equation; it is studied for instance in [39] where the authors obtain the Fourier transform of the solutions for any δ 2 ∈ (0, 1] expressed in terms of Mittag-Leffler functions and they give a representation of their inverse, in terms of stable densities; the special case δ 2 = 1/2 can be interpreted as a heat equation subject to a damping effect, represented by the 1/2-order time-derivative; in this case they show that the fundamental solution is the distribution of a telegraph process with Brownian time. In [42] the author investigates the existence and uniqueness of local (in time) solutions to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space . Some results about the well-posedness and regularity of solutions to (1.6) in bounded domains are presented for instance in [4,9,45].
Having in mind to apply the Fourier transform to the linear equation associated to (1.6), it is understandable as the problem of finding a suitable representation of solution is strictly related to solving fractional ordinary differential equations in the form for λ ∈ R. In [24] the authors develop the operational calculus of Mikusiński's type for the Caputo fractional differential equation, in order to obtain exact solutions of the initial value problem associated to (1.7) through Mittlag-Leffler type functions. The special cases δ 1 = 1, δ 2 ∈ (0, 1) and, respectively, δ 1 = 2, δ 2 ∈ (1, 2) have been deeply investigated in [17] taking λ = 1 and are referred as the composite fractional relaxation equation and, respectively, the composite fractional oscillation equation; here, by applying the technique of Laplace transforms they derive the analytical solutions to such equations. In particular, the fractional differential equation in (1.7) with δ 1 = 1, δ 2 = 1/2 corresponds to the Basset problem: it represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the force of gravity. The situation of a sphere subjected to gravity was first considered independently by Boussinesq [6] and by Basset [2], who introduced a special hydrodynamic force, which is nowadays referred to as Basset force. The whole was summarized by Basset himself in a later paper [3], and, in more recent times, by Hughes and Gilliand [19]. Nowadays the dynamics of impurities in unsteady flows is investigated as shown by several publications, which aim to provide more general expressions for the hydrodynamic forces, including the Basset force, in order to fit experimental data and numerical simulations, see e.g. [5, 23, 29-32, 37, 38]. For a complete history of the Basset problem one can refer to [7].
In [14] the Cauchy-type problem is investigated in the case β ∈ (1, 2). Under suitable assumptions on the nonhomogeneous term g(t, x), the authors investigate some L p − L q decay estimates for the solution w; then, they apply such estimates to study the corresponding semilinear problem.

Notation
In this paper, L ∞ t X denotes L ∞ ([0, ∞), X ), i.e., the space of essentially bounded functions from [0, ∞) to X . Moreover, C b ([0, ∞), X ) denotes the space of continuous bounded functions from [0, ∞) to X . In the both cases, g L ∞ t X denotes the norm sup t≥0 g(t, ·) X . For any s ∈ R, is the (fractional) Sobolev space equipped with norm f H s = ξ sf L 2 , where the symbol ξ denotes the quantity 1 + |ξ | 2 . For f ∈ H s with s ≥ 0, we also define For q ∈ (1, ∞) and s ∈ R we also define the Bessel potential space [1] We recall that f H s,q ≤ f L q for any s < 0.
In this paper, f g means that f ≤ Cg for some constant C > 0, and f ≈ g means that f g f .

Results
We first present a sufficient condition on g such that the solution to (1.1) remains bounded in H s .

Theorem 1
Let n ≥ 1 and s ∈ R. Assume that u 0 ∈ H s and that g ∈ L ∞ t H s−2+ε for some ε > 0. Then the solution u to (1.1) is in C([0, ∞), H s ) and for any t ≥ 0, where C > 0 and C ε > 0 are independent of t. In particular, u is in When n ≤ 3 and s ∈ [0, 2 − n/2), the embedding L 1 → H s−2 holds, hence the smoothing effect is sufficient to describe the asymptotic profile in H s of the solution to (1.1), in the form M K † Theorem 2 Let n = 1, 2, 3, and s ∈ [0, 2 − n/2). Assume that u 0 ∈ L 1 ∩ H s , and that g ∈ L ∞ t H s−2+ε for some ε > 0, also satisfies we also assume that g ∈ L ∞ t H s+a−4,q for some q ∈ (1, 2), where and that Then, u is in C b ([0, ∞), H s ) and there exists C > 0 independent of t, such that (B 2 = 0 in the following, if (1.10) does not hold) In particular, if lim sup t→∞ Q 1 (t) = 0 and, in addition, lim sup t→∞ Q 2 (t) = 0 when (1.10) holds, then the solution u also satisfies (1.14) When M = 0, we may say that the asymptotic profile of u(t, ·) in H s as t → ∞, is M K † 0 (t, ·). Remark 1 We stress that s − 4 < s − 2 + ε < 0 for sufficiently small ε > 0, and s + a − 4 < 0, in Theorem 2, so that B 1 and B 2 in (1.9) and (1.12) are finite if there exists B > 0 such that (1.15) in particular, conditions (1.9) and (1.12) are satisfied for B 1 = 2B and (1.16) We may apply Theorem 2 to study the semilinear problem where f (u) = |u| p for some p ≥ 2, or, more in general, Then, as a consequence of Theorem 2 we have the following result: Corollary 1 Let n = 1, 2 and assume that p ≥ 1 + 2/n in (1.18). Fix s such that Then there exists ε > 0 such that for any initial data Thus, when M = 0 and p > 1 + 2/n, Corollary 1 means that the nonlinearity does not influence the asymptotic profile of the solution to (1.17). The critical exponent 1 + 2/n is sharp [10]. We notice that in the critical case p = 1 + 2/n, Corollary 1 guarantees the existence of a global small data solution, but the asymptotic profile of the solution to (1.17) depends on the nonlinearity, in general.
Theorems 1 and 2 are based on the following representation formula for the solution to (1.5).

Lemma 1 Assume that y = y(t) solves the Cauchy problem (1.5).
Then where K 0 and K 1 have the following integral representations: The fact that K 0 and K 1 are different in Lemma 1 means that the Duhamel's principle does not hold in classical sense.

Remark 2
Applying the change of variable xt = τ 1 β we obtain the following representations for K 0 (t) and K 1 (t): dτ.

Comparison with the damped wave equation
By Theorem 2 we deduce that the solution u to the homogeneous Cauchy-type problem associated to (1.1), asymptotically behaves as the solution v to (1.3). There are several analogies with the diffusion phenomenon studied for the damped wave equation [27,34,36]: The asymptotic profile of the solution is described by 4t is the fundamental solution to the heat equation and under the assumption of nonzero moment condition M = 0. This diffusion phenomenon allowed to prove the global existence of small data solutions to the semilinear problem with power nonlinearity f (u), in the supercritical case p > 1 + 2/n (see [44]), as for the semilinear heat equation. A nonlinearity, in general, influences the asymptotic profile of the solution, see [21]. In Corollary 1, we showed that this is not the case for our equation in the supercritical case. This latter phenomenon is a consequence of the special structure of the Cauchy-type problem for fractional equations, and of the fact that the Duhamel's principle does not hold in classical sense, see Lemma 1.
Diffusion phenomena hold, more in general, for evolution equations when the damping is effective damping according to the classification introduced by the authors in [12], that is, 2θ < σ. Here, (−Δ) α denotes the fractional Laplace operator of order α > 0 defined on S as (−Δ) α f = F −1 (|ξ | 2αf ). If θ = 0 the solution to (1.23) behaves asymptotically like the solution to the corresponding diffusive equation, with initial data u 0 + u 1 , namely by e −t(−Δ) σ (u 0 + u 1 ); for θ > 0 a double diffusion phenomenon holds, that is, two different diffusive equations compete to describe the asymptotic profile of the solution to (1.23) (see [11]). On the other hand, when 2θ > σ, the asymptotic profile to (1.23) is completely different; in particular, the wave structure appears and oscillations come into play (see [20]). Inspired by the results just described, the main goal of the present paper is to show how the fractional in time derivative ∂ β t u in (1.1) influences the asymptotic profile of the solution with respect to the undamped heat equation: the presence of the fractional order member deeply influences the structure of the fundamental solution of equation (1.1); as a consequence, a dissipative-smoothing effect appears and the asymptotic profile of the solution to (1.1) is described by M K † 0 (t, ·), independently on the nonhomogeneous term, under suitable decay assumptions on g(t, ·) (see Theorem 2).

Proof of Lemma 1
In order to prove Lemma 1 we will use the Laplace transform method. Given a function ϕ = ϕ(t) of a real variable t ∈ R + = [0, ∞), L(ϕ) denotes its Laplace transform defined by Under suitable assumptions, the inverse Laplace transform of a given function F = F(s), holomorphic in some half-plane { s > λ}, is given for any t ∈ R + by the formula where a > λ.
The Laplace transform has many properties which are useful for studying dynamical systems. In particular, we mention that for any α ∈ (0, 1] the following transform rule holds for suitable good functions ϕ (see, for instance, [22]); such formula will allows us to transform the fractional differential equation in (1.5) in a functional equation. Let us apply the Laplace transform to the fractional differential equation in (1.5). Applying the identity (2.2) we get the functional equation Here and hereafter, for any s ∈ C, with s = re iθ , and α ∈ (0, 1) we are denoting by s α its root of order α on the principal branch, i.e., s α := r α e iαθ with θ ∈ (−π, π).
Thus, using the convolution theorem where for any t ≥ 0 we set We notice that thanks to the properties of the Laplace transform. The remaining part of this section is devoted to the proof of Lemma 1 starting from the identities in (2.4). In order to get this aim we will use two different approaches: the first approach is based on the direct evaluation of the inverse Laplace transforms in (2.4); on the other hand, in the second approach we will express K 0 (t) and K 1 (t) as a combination of Mittag-Leffler functions.

Laplace Transfrom method
In order to get the desired integral representations in (1.21)-(1.22) we will use the integral formula for the inverse Laplace transform given in (2.1).
Let us define the function ω : C → C such that ω(s) := s + s β + λ. We remark that ω has no zeros in C: suppose that s 0 = r 0 cos θ 0 + ir sin θ 0 is a zero of ω; then the couple (r 0 , θ 0 ) satisfies the system r cos θ + r β cos(βθ ) + λ = 0, r sin θ + r β sin(βθ ) = 0; (2.5) where a > 0 and F(t, s) := e st /ω(s) for any t ≥ 0. In order to calculate such integral, we consider the region delimited by the so called Hankel path, defined by the segment (a − ib, a + ib), arcs C R and C R , segments I and I I , and the circle C r as represented in Fig. 1 As a consequence of the Jordan lemma we immediately obtain that moreover, it is easy to check that also the integral over C r tends to 0 as r → 0. Therefore, we conclude We use the same approach to get the desired integral representation of K 0 , defined as where G(t, s) := e st (1+s β−1 )/ω(s). Also in this case, the path is contained in C\R − , where the function G(t, s) is holomorphic. Thus, using the same notation as for K 1 , we get As a consequence of the Jordan lemma we can conclude that the integrals over C R and C R tend to 0 as R → ∞; moreover, it is easy to check that also the integral over C r goes to 0 as r → 0. Thus, we get This complete the proof of Lemma 1.

Mittag-Leffler functions
In this section we show how the simplest case β = 1/2 can be treated with an alternative approach. In order to obtain the desired representations (1.21) and (1.22) we follow the idea given in [17] to provide the solutions K 0 and K 1 in terms of Mittag-Leffler functions. By (2.3) we know that the Laplace tranform of the solution y = y(t) to (1.5) has the representation where for any t ≥ 0, the values of K 0 (t) and K 1 (t) are defined in (2.4). Let s ± denote the two roots of the second degree polynomial s 2 + s + λ; then, it holds Here, s ± satisfies the useful relations Thus, we can write where A ± := ±s ± /(s + − s − ). As a consequence it is possible to write K 0 (t) and K 1 (t) as a linear combination of Mittag Leffler functions; such functions are defined by the following series representation, , α > 0, z ∈ C.
As a consequence of (2.8), by (2.6) and (2.7) we obtain and (2.10) In order to get the desired integral representations in (1.21)-(1.22) we will use the following useful lemma that is a particular case of Theorem 1 in [40].

Remark 3
In the general case β ∈ (0, 1) one can express the kernels K 0 and K 1 in terms of multivariate Mittag-Leffler functions (see [24]): Indeed, one can prove

Decay estimates
Applying the Fourier transform with respect to the space variable in (1.1) we get the following Cauchy problem for a parameter dependent fractional differential equation: (3.1) By Lemma 1, the solution iŝ As a consequence, we obtain the representation By the change of variable ξ → t β 2 ξ , for any s ≥ 0 and 1 ≤ q ≤ ∞, we obtain Noticing that it is useful to divide the half-line R + in two regions, depending on t and ξ : Therefore, we can estimate Thanks to (3.8), we prepare the pointwise estimates forR 0 (t, ξ) andR 1 (t, ξ).

Lemma 3
The following estimates hold: (3.10) Proof In order to get the desired estimates, we split the integral in the two regions I t,ξ and J t,ξ defined in (3.7). We first considerR 1 (t, ξ). By using estimate (3.8) in I t,ξ , we get In particular, since β < 1 this latter term may be estimated by the quantity t −(1−β) t 1−β |ξ | 2 −M , for any M ≥ 0. However, for short times, the estimate above is singular at t = 0, so we proceed in a different way: where we first used the change of variable s = τ 1 β − t 1−β |ξ | 2 and then the change of variable r = s/(t 1−β τ 0 ). Summarizing, we proved for any M ≥ 0. We now consider the integral over J t,ξ . If we use (3.8) to estimate ϕ(t, τ, ξ) t 2(1−β) |ξ | 4 , then we find whereas, if we use (3.8) to estimate ϕ(τ, ξ ) ≥ t 2(1−β) τ 2 as we did in I t,ξ , we find Therefore, we obtain Comparing with the estimate of the integral over I t,ξ , we conclude the proof of (3.9) for t ≥ 1, using t 1−β 2 ξ −4 ≤ ξ −4 . However, for short times, the estimate above is singular at t = 0, then we estimate where we used first the change of variable s = τ 1 β −1 and then the change of variable r = s/t 1−β . Summarizing, at short time, we may estimate Comparing with the estimate of the integral over I t,ξ , we conclude the proof of (3.9) for t ≤ 1, using t . We now prove (3.10). If we use (3.8) to estimate ϕ(t, τ, ξ) t 2(1−β) |ξ | 4 when we integrate over J t,ξ , then we find On the other hand, if we use (3.8) to estimate ϕ(t, τ, ξ) t 2(1−β) (|ξ | 4 + τ 2 ), we obtain Therefore, Now we consider the integral over I t,ξ . In this case, we get where we used first the change of variable s = τ 1 β − t 1−β |ξ | 2 and then the change of variable r = s/(t 1−β τ 0 ). Therefore, for any M ≥ 0. Comparing with the estimate over J t,ξ , we conclude the proof of (3.10).
As a straightforward consequence of (3.5) and of Lemma 3, we have the following estimates.

13)
provided that s + n/q < 4, and We stress that since (3.11) holds for any s ∈ [0, 4], we may also write it in the form for any s ∈ [0, 4].
Proof For any ξ ∈ R n and s ∈ [0, 4], we use (3.9) to obtain On the other hand, if t ≥ 1, we get provided that s + n/q < 4. Estimate (3.12) immediately follows from (3.10). Moreover, if t ≥ 1, we get provided that s + n/q < 2.

Proof of the main results
Proof of Theorem 1 Using (3.4), we may estimate in order to estimate the last term, we split the integral in the two domains [0, (t − 1) + ] and [(t − 1) + , t]. Then, we apply estimate (3.15) in the first interval and (3.16) in the second one: This proves (1.8). The second part of the statement follows noticing that is bounded with respect to t, since β ∈ (0, 1).
The proof of Theorem 2 will be given throughout the following two main lemmas.

4)
for t ≥ 1, where C is a positive constant which does not depend on t.
Proof We first note that it holds In particular, we havê By using (3.8), on the one hand, being t 1−β |ξ | 2 ∼ τ 1 β in I t,ξ for any t ≥ 1 we can estimate for some p β > −1; on the other hand, for any t ≥ 1 we have for some q β > −1.
As a consequence, we can easily conclude since n < 4 − 2s. The proof of the desired result follows by identity (4.5).

Proof of Theorem 2
The solution to (1.1) is in C b ([0, ∞), H s ) thanks to Theorem 1. By using (3.4), on the one hand, we may estimate here, we used (3.12) for t ≤ 1 and (3.14), together with Plancherel and Riemann-Lebesgue theorem, for t > 1; on the other hand, we have Recalling that s < 2 − n/2, we use Plancherel and Riemann-Lebesgue theorem together with Lemmas 5 and 6 to conclude that It remains to estimate the integral term in (4.6) and (4.7): let t ≤ T , with T > 0 arbitrarily large; then, as a consequence of assumption (1.12) and estimate (3.11) we obtain t 0 |ξ | sK Let us consider now t > T ; we separately estimate three integrals: Due to s < 2, using (3.15), we may estimate Similarly, if we assume that then, using (3.15), we may estimate If (4.10) does not hold, that is, (1.10) holds, using (3.15) we may estimate where a is as in (1.11). By the Hardy-Littlewoow-Sobolev theorem, provided that q ∈ (1, 2), where a is as in (1.11). Therefore, We obtain We now prove Corollary 1.

Proof of Corollary 1
We equip the evolution space and we define the operator where We will prove the existence of the unique global (in time) solution to (1.17) as the fixed point of the operator N . Hence, in order to get the global (in time) existence and uniqueness of the solution in X (T ), we need to prove the following two crucial estimates: (4.14) with C > 0, independent of T . As a consequence of Banach's fixed point theorem, the conditions (4.14) and (4.15) guarantee the existence of a uniquely determined solution u to (1.17). We simultaneously gain a local and a global (in time) existence result.
Indeed, let R > 0 be such that C R p−1 < 1/2. Then N is a contraction on X R (T ) = {u ∈ X (T ) : u X (T ) ≤ R}, thanks to (4.15). The solution to (1.17) is a fixed point for N , so if K 0 (t, ·) * (x) u 0 X (T ) ≤ R/2, then u ∈ X R (T ), thanks to (4.14). As a consequence, the uniqueness and existence of the solution in X R (T ) follows by the Banach fixed point theorem on contractions. The condition K 0 (t, ·) * (x) u 0 X (T ) ≤ R/2 is obtained taking initial data verifying u 0 L 1 ∩H s ≤ ε, with ε such that Cε ≤ R/2. Since C, R and ε do not depend on T , the solution is global (in time).
In particular, if p > 1 + 2/n we have and then, we prove (1.20) as a consequence of (1.14).

Conclusions
In Theorem 1, we provided sufficient conditions for the non-homogeneous term g(t, x) which allow to obtain the boundness of the solution to (1.1) in H s ; then, we describe its asymptotic profile in Theorem 2, showing that it is independent of g, provided that g(t, ·) satisfies some additional decay assumptions. This latter effect is related to the special structure of the solution to the Cauchy-type problem for equations with Caputo fractional derivatives, in relation to the non-homogeneous term g(t, x). There exist many functions g(t, ·) which satisfy the desired conditions. Let us test the assumptions of Theorem 1 for the special class of auto-similar g: g(t, x) = t γ h(t 2γ n x), for some γ > 0 and for any t > 0.
One may proceed with similar reasoning to test the assumptions of Theorem 2 for a self-similar g.
One could also investigate the possibility to take g(t, ·) in different functional spaces; for instance, following the proof of Theorem 1 one could also prove that u is in C b ([0, ∞), H s ) and Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.