Nonexistence Results for Some Classes of Nonlinear Fractional Differential Inequalities

We study the nonexistence of global solutions for new classes of nonlinear fractional di ﬀ erential inequalities. Namely, su ﬃ cient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

Due to the importance of fractional calculus in applications (see e.g. [1][2][3][4][5]), in the past few decades, there has been a growing interest in the study of fractional differential equations. In particular, from the theoretical point of view, the existence of solutions for different classes of fractional differential equations was investigated in many contributions (see e.g. [6][7][8][9][10][11][12] and the references therein).
It was shown that, if then problem (4) does not admit nontrivial global solution.
In [15], Furati and Kirane investigated the system of nonlinear fractional differential equations to the initial conditions where 0 < α, β < 1, p, q > 1, and u 0 , v 0 > 0. It was shown that, if then solutions to system (6) subject to (7) blow up in a finite time.
For the issue of nonexistence of global solutions for fractional in time evolution equations, we refer to [6,[23][24][25] and the references therein.
On the other hand, to the best of our knowledge, the nonexistence of global solutions for problems of types (1) and (2) was not yet investigated.
Before stating our main results, let us mention what we mean by global solutions to problems (1) and (2). Definition 1. A function u ∈ AC 2 ð½0,∞ÞÞ is said to be a global solution to problem (1), if u satisfies for almost every where t > 0, and Definition 2. A function u ∈ AC 2 ð½0,∞ÞÞ is said to be a global solution to problem (2), if u satisfies for almost every where t > 0, and We first consider problem (1). We discuss separately the cases u 1 > 0 and u 1 = 0. Theorem 3. Let α, β, γ ∈ ð0, 1Þ, and u 0 ∈ ℝ . If u 1 > 0, then for all p > 1, problem (1) admits no global solution.
(i) If γ ≤ α, then for all p > 1, the only global solution to problem (1) is u ≡ u 0 (ii) If γ > α, then for all the only global solution to problem (1) is u ≡ u 0 .
then problem (2) admits no global solution.
We discuss below some special cases of Theorem 6.
where a ∈ ℝ and a ≠ 0.

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problem (2) admits no global solution.
The rest of the paper is organized as follows. In Section 2, we recall briefly some standard notions on fractional calculus and prove some properties. Section 3 is devoted to the Proofs of Theorems 3, 4, 5, and 6 and Corollaries 7 and 8.

Some Preliminaries
We denote by ACð½0,∞ÞÞ the space of absolutely continuous functions on ½0, ∞Þ. Given an integer n ≥ 2, we denote by A C n ð½0,∞ÞÞ the space of functions f which have continuous derivatives up to order n − 1 on ½0, ∞Þ such that f ðn−1Þ ∈ A Cð½0,∞ÞÞ. Here, f ðn−1Þ denotes the derivative of order n − 1 of f .

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Using the change of variable z = ðs − t/T − tÞ, one obtains where Bð·, · Þ is the beta function. Using the property one obtains (25).

Proofs
The proofs of our results are based on the test function method (see e.g. [26])and some integral estimates.
Proof of Theorem 3. Let us suppose that u ∈ AC 2 ð½0,∞ÞÞ is a global solution to (1). For T > 0, multiplying the differential inequality in (1) by ξ, where ξ is the function defined by (24), and integrating over ð0, TÞ, one obtains Without restriction of the generality, we may suppose that On the other hand, using Lemma 10, one obtains Using an integration by parts, the initial conditions and (25), it holds that On the other hand, by Lemma 9 and using the initial conditions, one obtains Therefore, by (35), one obtains Using an integration by parts, the initial conditions, (26) and Lemma 10, it holds that Similarly, one has Next, using (32), (38), and (39), one obtains On the other hand, using ε-Young inequality with 0 < ε < ð1/2Þ, one obtains Journal of Function Spaces where Cðε, pÞ is a positive real number that depends only on ε and p. Similarly, one has Hence, it follows from (40), (41), and (42) that Since 0 < ε < ð1/2Þ, one deduces from (43) that On the other hand, by (25), one has and which yield Since u 1 > 0, one deduces that where C 1 = ðΓðλ + 1Þ/Γð2 + λ − αÞÞ > 0. Next, using (28) with ρ = γ and κ = α, one obtains which yields where Similarly, one has where Therefore, it follows from (44), (48), (50), and (52) that which yields Notice that for all p > 1, one has

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Hence, using (56) and passing to the limit as T ⟶ +∞ in (55), one obtains u 1 ≤ 0, which contradicts the fact that u 1 > 0. Therefore, one deduces that for all p > 1, problem (1) admits no global solution.