ON THE INITIAL VALUE PROBLEM FOR FRACTIONAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH A CAPUTO - FABRIZIO DERIVATIVE ∗

. In this paper, a time-fractional integrodiﬀerential equation with the Caputo - Fabrizio type derivative will be considered. The Banach ﬁxed point theorem is the main tool used to extend the results of a recent paper of N.H. Tuan and Y. Zhou [1]. In the case of a globally Lipschitz source terms, thanks to the L p − L q estimate method, we establish global in time well-posed results for mild solution. For the case of locally Lipschitz terms, we present existence and uniqueness results. Also, we show that our solution will blow up at a ﬁnite time. Finally, we present some numerical examples to illustrate the regularity and continuation of the solution based on the time variable.


Introduction
Fractional calculus has many applications in mechanic, physics and engineering science... For example, a fractional diffusion equation is a generalization of a classical diffusion equation which models anomalous diffusive phenomena. For results on fractional derivatives like Riemann -Liouville type or Caputo type we refer the reader to [24][25][26][37][38][39] and the references therein. Recently Caputo and Fabrizio [35] introduced a new fractional derivative, which called Caputo -Fabrizio fractional derivative.
Our paper focuses on studying the existence and uniqueness of a mild solution of time-fractional Volterra integrodifferential equations with the new type of fractional derivative. Let T be a positive number and α ∈ (0, 1), we consider the following intial value problem here, u 0 is the initial data, the functions G, ϕ and the domain D will be defined later. The symbol CF D α t stands for the Caputo -Fabrizio type time fractional derivative operator of order α (see [1,14]). We recall the definition of the Caputo -Fabrizio type time fractional derivative as follows. Let a > 0, 0 ≤ α ≤ 1, and for a function w belongs to H 1 (0, a), its Caputo-Fabrizio fractional derivative is defined as (see [14]) where M is a normalization function such that M (0) = M (1) = 1. When ϕ = 0, the authors in [1] investigated the well-posedness of an initial value problem for a fractional diffusion equation. In [16], the existence of the solution to an initial value problem for a linear differential equation was investigated and the authors applied their results to the mass-spring-damper motion in the general case. We refer the reader to [31] for results on the existence of the Korteweg-de Vries-Burgers equation with a fractional Caputo -Fabrizio derivative. For more results on time fractional partial derivative equations topic, we refer the reader to [2][3][4][5][6][7][8][9][10][11][12][13]. When (ϕ = 0), Problem (1.1) can be used to model some natural phenomena with memory effects. In [32] the authors considered memory effects on the dynamics of non-Newtonian fluids and viscoelastic models for the dynamics of turbulence statistics in Newtonian fluids on the modified 3D Navier -Stokes equation x ∈ Ω, (1. 3) In [33] the authors compared the energy dissipation produced by the internal motion and the memory effect i.e they studied the behavior between the equation u tt + ∆ 2 u + α∆u t = 0 in Ω (1.4) and the equation We also refer the reader to [17], [18], [19], [40] and the references therein. In [17] the authors established the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions of the equation which arises in the Coleman-Gurtin theory of heat conduction with hereditary memory. However, to the best of our knowledge, there are few results on Volterra diffusion equations with the Caputo -Fabrizio derivative (Problem (1.1)). Our first main goal is to present global results on the whole space R N under a global Lipschitz condition on the source term. The second goal is to prove the existence and uniqueness of mild solutions and present a blow-up alternative for mild solutions of Problem (1.1) under a locally Lipschitz condition on G and ϕ.
The paper is organized as follows. In Section 2, we introduce some important vector spaces, Lemma's, preliminaries, and the definition of mild solutions of (1.1). In Section 3, we prove global existence for the mild solution of Problem (1.1) when D = R N , with globally Lipschitz source terms. Finally, in the last Section, a local well-posed result will be given for Problem (1.1) with locally Lipschitz condition when D is an open bounded domain with smooth boundary in R N .

Preliminary material
Assume that D is a bounded domain whose boundary is sufficiently smooth. We consider the following spectral problem for the Laplace operator ∆ which admits a family of eigenvalues 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · ≤ λ j · · · ∞ and a corresponding family of eigenvectors {ψ j } j∈N .
Let (B, · B ) be a Banach space, then, we denote by L p (0, T ; B) the Banach space of measurable functions f : (0, T ) → B eqquiped with the norm The space of all k time's derivative continuous functions C k ([0, T ]; B), k ∈ N is a Banach space with the norm We denote by X h,µ ((0, T ]; B), the subspace of all functions f in C([0, T ]; B) such that Assume that D is a bounded domain whose boundary is sufficiently smooth. For any σ > 0, we define the fractional Hilbert scale space by which is the duality space of H σ (D). The notation ·, · * denotes the dual inner product between H −σ (D) and H σ (D).
Lemma 2.1. (see [14]) The Laplace transform of the Caputo -Fabrizio derivative is given by Lemma 2.3. (see [15])Assume that D ≡ R N . Let 1 ≤ p ≤ q and w ∈ L p (D) ∩ L q (D). Then, there exists a constant C such that 3. Result on the whole space R N with globally Lipschitz source term In this section, we consider the existence and uniqueness of mild solution for Problem (1.1) when D ≡ R N under the following assumptions for the source terms and where L is not dependent on a, b.
Proof. The proof begins by defining a mapping J : X h,µ ((0, T ]; Z) −→ X h,µ ((0, T ]; Z) as follows We first use Lemma 2.8 and the inequality e −r ≤ c θ r −θ (r > 0, 0 < θ < 1) to obtain Next, we will deal with the remain term Jw(t) − AT (αt)u 0 . To this end, for any w, v ∈ X h,µ ((0, T ]; Z), we will consider the following equation where Using the same calculation ideas as in (3.5), we see that (3.10) Then, it follows that The above estimate gives us the one below In Lemma 2.8, taking p = q and setting These two estimates above lead us to the following result (3.14) By similar arguments, we also obtain On account of the results stated above, by choosing v = 0, it's easily check that if we take w in X h,µ ((0, T ]; Z), Jw will be in X h,µ ((0, T ]; Z). In addition, the standard smooth effect of the semigroup T (t) will ensure the continuity of J on (0, T ]. Therefore, we conclude that, J is invariant on X h,µ ((0, T ]; Z).
In the other hand, from the assumptions on p, q, h, we can easily deduce that Combining (3.7), (3.14), (3.15), we have From (3.16), there exists a sufficiently large µ 0 such that Consequently, J is a contraction on X h,µ ((0, T ]; Z). Now we conclude that Problem (1.1) has a unique solution in X h,µ ((0, T ]; Z) and the proof is completed.

Problem (1.1) under locally Lipschitz source term
Throughout this section, we suppose that D ⊂ R N is an open bounded domain with smooth boundary and our solution will vanish on the boundary of D. Furthermore, we consider the following assumptions for the source terms.
Let ν < σ < ν + 2 and m, n > 1. Suppose that for any a, b ∈ R, functions G, ϕ satisfy To deal with the initial value problem with bounded domain, we will rewrite (2.7) in the form of Fourier series as follows (see [1]) and

Local existence and uniqueness of the mild solution
Proof. Using the triangle inequality, we have Thanks to the Parseval identity, the following estimate holds The above result gives Similarly, the second conclusion of this lemma can also be drawn Proof. Set u 0 = u0 1+λj (1−α) and we can check immediately that u 0 ∈ H σ (D). Next, for a fixed positive constantK, we consider the following space and define the operator J : A → A by (4.10) Note that A is a complete space with respect to the usual sup norm. For the purpose of using the Banach fixed point theorem, we first need to show that Jw ∈ C([0, T 0 ]; H σ (D)) for any w ∈ C([0, T 0 ]; H σ (D)). Let 0 < t ≤ t + ε ≤ T 0 . Using the inequality |e −a − e −b | ≤ |a − b|, for a, b > 0, we have (4.11) In the other hand, it follows from Lemma 4.1 that (4.13) By similar arguments, we also obtain Hence, if w belongs to C([0, T 0 ]; H σ (D)), then, Jw is in C([0, T 0 ]; H σ (D)). Next, we set and choose a sufficiently small T 0 such that Using the inequality 1 − e r ≤ r, we have Let w ∈ A. From Lemma 4.1 and the above estimate, we get Thus, Jw ∈ A. Now, we show that J is a contraction. Let v, w ∈ A, for any t ∈ [0, T 0 ], we have dz.

(4.18)
Note for the first term (one can apply a similar argument for the second term) From this point of view, we can conclude that J is a contraction on A. Therefore, there exists a unique solution u of (4.10) and the theorem is proved.

Continuation and blow-up alternative
In this subsection, we present a continuation result and a blow-up alternative for the mild solution of Problem (1.1). Proof. Consider the following space It's obvious that B is a complete space. Define a mapping J : dz.
Using some arguments as in the proof of the previous theorem, we have Claim 1. (4.23) Combining the above claims, we can choose some h > 0 and M > 0 such that the right hand side of the above estimate is less than M, so J v is in B.
Next, take v, w ∈ B. Then We can find a C h < 1 (depending on h) such that Then J is a contraction on B. Now apply the Banach Fixed Point Theorem to get the desired result. Proof. Suppose that T max < ∞ and there exists a constant > 0 such that Consider a sequence of positive numbers {t k } k∈N ⊂ [0, T max ) such that t k k→∞ − −−− → T − max and the sequence {u(t k )} is a subset of H σ (D). We show that {u(t k )} k∈N is a Cauchy sequence in the Banach space H σ (D). Let t m , t n ∈ H σ (D), and without loss of generality suppose that t m < t n . Then, we have (4.32) Using the inequality |e −a − e −b | ≤ |a − b|, we have From the same method as in the proof of Lemma 4.1 we have Next, we have Finally note From the above estimates we have Hence, {u(t k )} k∈N is a Cauchy sequence in H σ (D). We deduce that {u(t k )} k∈N has a limit u Tmax in H σ (D). Because {t k } k∈N is arbitrary, we have the following limit (4.39) Therefore, we can define u over [0, T max ]. By applying Theorem 4.3, the mild solution of Problem (1.1) on [0, T max ] can be extended to some larger interval and this leads to a contradiction with the definition of T max . The result follows.

Numerical example
In this section, we consider some numerical examples to illustrate two properties of the mild solution in the Fourier form (4), namely the regularity and continuation of the solution u based on the time variable.
Choose T = 1 and the domain D = [0, π]. We focus on the equation with the intial condition Then we have the orthogonal basis in L 2 (0, π) is ψ j (x) = 2 π sin(jx) and the corresponding eigenvalues {λ j } = {j 2 | j = 1, 2, ...}. The spatial variable are discretized as follows , for l = 0, 1, ..., X l + 1, where X l > 0 is a given integer number which is the number of partitions. In addition, the scalar product of g and h in L 2 (0, π) is given by g, h L 2 (0,π) = π 0 gh dx.
According to (4), we have the solution as follows u(x, t) where Next, to consider the regularity of the solution based on the time variable, we focus on the absolute error estimation between the solution at t and the solution at t * for some values of α as follows Numerical results are presented in Tables 1, 2, 3, 4 and Figures 1, 2, and 3, i.e., we show the solutions in the 2D and 3D graph for t ∈ {0.2, 0.4, 0.6, 0.8}, α ∈ {0.1, 0.5, 0.9}, respectively. From the above results, it can be concluded that the solution u at t * approaches the solution u at t when t * tends to t. In addition, we also show a 3D graph of the solution u for different values of α in {0.1, 0.5, 0.9}, see Figure 4.

Conclusion
This paper considers the initial value problem for a time-fractional equation with the Caputo -Fabrizio derivative and memory effect on the source term. In our work, the formula of the mild solution is given and based on it, we investigate the global in time results when the source terms are globally Lipschitz and when the source terms are locally Lipschitz, the local in time well-posedness and finite-time blow-up results are obtained. ...