Sufﬁcient Criteria for the Absence of Global Solutions for an Inhomogeneous System of Fractional Differential Equations

: A nonlinear inhomogeneous system of fractional differential equations is investigated. Namely, sufﬁcient criteria are obtained so that the considered system has no global solutions. Furthermore, an example is provided to show the effect of the inhomogeneous terms on the blow-up of solutions. Our results are extensions of those obtained by Furati and Kirane (2008) in the homogeneous case.

The study of the absence of global solutions for differential equations (or fractional differential equations) furnishes important indications on limiting behaviors of many physical systems. In industry, the knowledge of the finite time blow-up can prevent accidents and malfunctions. It can also be useful for the improvement of the performance of machines and the extension of their life-span.
In [20], Furati and Kirane investigated the system which is a special case of Equation (1) with z i ≡ 0, i = 1, 2. They proved that, if x 0 , y 0 > 0 and then Equation (2) has no global solutions. Next, Kirane and Malik [14] studied the profile of the blowing up solutions of Equation (2). Let us mention that several results related to the finite time blow-up of solutions of fractional differential equations were obtained in previous contributions (see e.g., [13,[21][22][23][24][25] and references therein). However, systems of the type seen in Equation (1) were not studied previously. Here, our aim is to study the effect of the inhomogenous terms z i (t), i = 1, 2, on the blow-up of solutions of Equation (2).
Before presenting the main results, we first recall briefly certain standard notions on fractional calculus that will be used throughout this paper. For more details, see e.g., [9].
Let ξ > 0. The fractional integrals of order ξ of a function η ∈ C([0, µ]), µ > 0, are given by Let where κ ≥ 0. Then Suppose now that 0 < ξ < 1. The derivative of order ξ of a function η ∈ C 1 ([0, µ]) in the Caputo sense, is given by The pair of functions (x, y), where x, y ∈ C 1 ([0, ∞)), is a global solution of Equation (1), if it satisfies Equation (1) for all t > 0. Let us recall that the system of Equation (1) is investigated under the assumptions: Now, we present our results.
then Equation (1) has no global solutions.
The following example shows the effect of the inhomogeneous terms z i (t), i = 1, 2, on the blow-up of solutions of Equation (2) Example 1. Consider the system where t > 0, x 0 ≥ 0,y 0 ≥ 0, p > 1, q > 1 and for all t ≥ 0. One observes easily that Furthermore, it follows from Equation (9) that Hence by Theorem 1, one deduces that under condition Equation (9), for all p, q > 1, Equation (8) has no global solutions.

Proof of the Main Result
The proof is based on the nonlinear capacity method (see e.g., [26]). More precisely, we first suppose that (x, y) is a global solution of Equation (1). Next, we multiply both equations in Equation (1) by an adequate test function that depends of a parameter T 1, and we integrate by parts over the interval (0, T). Using standard integral inequalities, the condition in Equation (6) and passing to the limit as T → ∞, a contradiction follows.
The detailed proof of Theorem 1 is given below.
Proof. We follow the steps mentioned previously.
Step 1 (Multiplication of both equations in Equation (1) by an adequate test function that depends of a parameter T 1): Suppose (x, y) is solution of (1) which is global. For κ, T 1, let After multiplication of the first equation in Equation (1) by ν(t) and integration over (0, T), we obtain Integrating by parts, we have Since ν(T) = 0 and ν(0) = 1, we get Using Equations (5) and (3), we have Again, we integrate by parts, we obtain On the other hand, by Equation (4), one has I 1−α T ν (T) = 0. Therefore, it holds that Further, using Equation (11)-(13), we deduce that Since x(0) ≥ 0 and I 1−α T ν (0) ≥ 0 by Equation (4), it holds that Similarly, after multiplication of the second equation in Equation (1) by ν(t) and integration over (0, T), using that y(0) ≥ 0 and I Next, we set We use Hölder's inequality to get and Similarly, one has and T 0 |y| dI Next, setting using Equations (14), (16), and (17), one deduces that Similarly, setting using Equations (15), (18), and (19), one deduces that Consider now the case lim sup it follows from Equation (21) that The above inequality with Equation (20) yields Step 2 (Estimates and conclusion): Further, we shall estimate the term in right-hand side of the above inequality. Using the inequality Similarly, using Equations (4) and (10), we obtain Hence, using Equations (24)-(26), one deduces easily that where C > 0 is a constant. The above inequality with Equation (23) yield Next, using Young's inequality where C = pq−1 pq C pq pq−1 . On the other hand, by Equation (10), one has The above inequality with (28) yields Using similar argument as above, one obtains for some constant C > 0, which contradicts (29).
Author Contributions: All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.