On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefﬁcient

: This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i ∂ ω + ( 0, ∞ ) × R N , where N ≥ 1, ξ ∈ C \{ 0 } and p > 1, under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefﬁcient functions a 1 , a 2 ∈ L 1 loc ([ 0, ∞ ) , R ) , and provide two illustrative examples.


Introduction and Preliminaries
In this paper, we study the following initial value problem for the fractional in time nonlinear Schrödinger equation i α ∂ α t ω(t, z) + a 1 (t)∆ω(t, z) + i α a 2 (t)ω(t, z) = ξ|ω(t, z)| p , (t, z) ∈ (0, ∞) × R N , (1) where N ≥ 1, ξ ∈ C\{0} and p > 1, under the assumption that In the left-hand side of (1), i α = e i απ 2 , and ∂ α t means the Caputo fractional derivative in time of order α ∈ (0, 1). Later on, we need some regularities of the coefficient functions: a 1 , a 2 ∈ L 1 loc ([0, ∞), R), a 1 ≡ 0 (that is, not identically zero), and ∈ L 1 loc (R N , C). Schrödinger type equations arise naturally in the analysis of dispersive equations on large domains (for example, we refer to oceanic water waves). In addition, they are useful in the study of wave turbolence (as an application of statistical physics), see the comprehensive paper of Buckmaster-Germain-Hani-Shatah [1]. A significant topic for nonlinear partial Schrödinger equations is to establish sufficient conditions for the existence of solutions providing a localized behavior. Following, this feature, Rego-Monteiro [2] proved the existence of a traveling-wave solution, with solitary-wave behavior. Furthermore, also relevant to this study is the focus of qualitative research in symmetric domains to provide the symmetries of solutions (mainly, working with kinematical and dynamical algebras of Schrödinger type equations). A combination of suitable iteration methods, maximum principle and method of moving planes, is useful to detect symmetries of positive solutions and nonexistence results (see, for example, [3]). We also mention the recent works of Peng-Zhao [4] (global existence and blow-up of solutions) and Hoshino [5] (asymptotic behavior of solutions).
In [6], the authors considered the nonlinear Schrödinger equation in the form (that is, set α = 1 in (1)), in the context of optical soliton systems, where a 1 (t) plays the role of dispersion parameter, and a 2 (t) means the absorption coefficient.
In absence of absorption (that is, a 2 ≡ 0), and setting the dispersion term equal to one (that is, a 1 ≡ 1), Ikeda-Wakasugi [7] studied the mathematical model In detail, the authors considered the global behavior of solutions to problem (3), then they established a finite-time blow-up result of an L 2 -solution whenever p ∈ (1, 1 + 2 N ). The same problem (3) was discussed by Ikeda-Inui [8]. This time, they established a small data blow-up result of H 1 -solution, whenever p ∈ (1, 1 + 4 N ). Furthermore, a revival of interest to the study of Schrödinger equation is linked to the theory of fractional calculus (see, for example, the books of Kilbas-Srivastava-Trujillo [9] and Samko-Kilbas-Marichev [10]).
Let h ∈ C([0, T]) be a real-valued function and, as usual, denote by Γ(·) the Gamma function. Here, we recall that the Riemann-Liouville fractional integrals of order σ > 0, are given as We note that the limit of (I σ 0 h)(t), as t approaches zero from the right, is zero. So, we can put (I σ 0 h)(0) = 0 to extend by continuity I σ 0 h to [0, T]. The similar extended continuity holds for I σ T h, by taking (I σ T h)(T) = 0. In addition, the Caputo derivative of order σ ∈ (0, 1) of h ∈ C 1 ([0, T]) is obtained as In such a framework setting, we mention that the fractional version of (3), that is, received the attention of Zhang-Sun-Li [11], whose studies lead to nonexistence (blow-up) results of global solutions with suitable initial values and p ∈ (1, 1 + 2 N ). For further interesting contributions to the analysis of the blow-up behavior of solutions to fractional nonlinear Schrödinger problems, we mention the papers of Fino-Dannawi-Kirane [12] (semilinear equation with fractional Laplacian), Ionescu-Pusateri [13] (equation in dimension one with cubic nonlinearities) and Kirane-Nabti [14] (nonlocal in time equation). Finally, we recall the paper of Li-Ding-Xu [15] where a cubic non-polynomial spline method is implemented to solve the time-fractional nonlinear Schrödinger equation. Furthermore, the stability of the method is analyzed by Fourier analysis. Moreover, Shi-Ma-Ding [16] studied a fourth-order quasi-compact conservative difference scheme and provided precise informations on its stability. Resuming, now-a-day nonlinear Schödinger equations play a crucial role in modelling and controlling the behavior of optical soliton systems. The physical significance of considering the fractional in time version of such an equation, is mainly related to the description of the evolution of the above system in terms of Lévy motion, instead of the Brownian motion (see, for example, [17]).
Mathematically, we are concerned with the solvability of problem (1) and (2), depending on the behaviors of a 1 and a 2 at infinity. The approach is based on the nonlinear capacity method of Pohozaev [18], whose main skill is the ability to use specific test and cut-off functions related to the particular form of the nonlinear operator in the differential equation driving the problem. Following [18], and particularizing the method for Equation (1), we construct a nonexistence theorem and discuss some consequences, over the following definition.
In Definition 1, we made use of the following integration by parts rule:
This theorem is convenient in many cases of practical interest. Consequently, we discuss in details some special classes of coefficient functions in (1), and provide two illustrative examples.

Remark 2.
We point out that the constant choices a 1 ≡ 1 and a 2 ≡ 0, lead to interpret Corollary 1 as the nonexistence result of Zhang-Sun-Li ([11], Theorem 2.2).
So, we focus particular attention on the following cases: Case 1: q satisfies the inequality The choices in (8) lead us to obtain that for some C q,p > 0 (that is, a constant depending on q and p). This means that as T goes to infinity, which gives us hypothesis (P 1 ) (by the choice of α in (8)).
Case 2: q is restricted to positive values satisfying the inequalities Since by (8) we work with α ∈ (0, 2 N+2 ), then we have {q : (9) holds true} = ∅, , as T goes to infinity, which gives us again hypothesis (P 1 ) . In both the cases, by Corollary 1 we can conclude that there is no global weak solution to problems (1) and (2).
Summarizing the above facts, we have the following result. Relaxing the hypothesis on a 1 (that is, considering again a 1 ∈ L 1 loc ([0, ∞), R), a 1 ≡ 0), we set a 2 ≡ 0 in (1). This means that we reduce hypothesis (P 1 ) to the following So, Theorem 1 is restated in the following form. Corollary 3. If (P 1 ) and (P 2 ) a (or (P 2 ) b ) hold, then problems (1) and (2), with a 2 ≡ 0, admits no global weak solution.

Proof of Theorem 1
In this section we give the complete proof of Theorem 1. To construct the nonexistence result by contradiction, we assume that ω ∈ L p loc ((0, ∞) × R N , C) is a global weak solution to (1) and (2). Then, we focus on some characteristic truncation and comparison functions required by the Pohozaev nonlinear capacity method in [18], and distinguish two cases.
Proof. (P 1 ) and (P 2 ) a hold. Hypothesis (P 2 ) a , gives us ξ 1 = 0. So, involving (5) for all T > 0 and ϕ ∈ C 2 (Q T , R), ϕ ≥ 0, supp z ϕ ⊂⊂ R N , we get the equation which yields the following inequality Q T |ω| p ϕ dz dt + 1 Looking at the right-hand side of (17), together with the ε-Young inequality, choosing ε ∈ (0, 1 3 ) and a suitable C > 0 changing value from line to line (but not depending on T), we have Combining (18), (19) and (20) in the inequality (17), we get where we use the following notation, to compact the formula: To continue the proof we use a suitable cut-off function f ∈ C ∞ ([0, ∞)) assuming values in the interval [0, 1] with Now, we work with the C 2 (Q T , R)-function x(t, z) given as where Here, we need 1 with ρ > 0 to be chosen opportunely.
Since x ≥ 0 and supp z x ⊂⊂ R N , then we can set ϕ = x in (21) to obtain where C ,α = (1 + − α)C ,α . Using appropriate changes of variables and some manipulations, we get The definition of x (see (22)) leads us to Therefore, one obtains Now, we are going to estimate C x (T). Again, using (22), we get The combined effects of (23), (25), (26), (27) and (28) give us If we choose ρ = α 2 , then the above inequality reduces to Finally, we pass to the infimum limit as T goes to infinity in (29), use hypothesis (P 1 ), (recall 1 < p < 1 + 2 N , ∈ L 1 (R N , C)), and the dominated convergence theorem, then we have which contradicts hypothesis (P 2 ) a .
Proof. (P 1 ) and (P 2 ) b hold. We note that ω ∈ L p loc ((0, ∞) × R N , C) is a global weak solution to (1) and (2), ⇒ ν = −iω is a global weak solution to (6) and (7) (by Remark 1-(ii)). Now, the existence of no global weak solution to auxiliary problem (6) and (7) can be established by contradiction, on the same lines of the proof of previous Case 1, with hypothesis (P 2 ) a in the form Finally, we point out that We conclude that ( P 2 ) a and (P 2 ) b are equivalent.
In both the cases, we can exclude the existence of a global weak solution to problems (1) and (2).

Conclusions
The interest for nonlinear Schrödinger equations is dictated by various applications in physics. Two important directions of research are aimed to prove: -Existence and nonexistence (blow-up) results of global weak solutions. -Improve the analysis of the regularity and asymptotic behavior of solutions.
The existing literature provides some basic approaches and other useful tools to develop a full analysis of the dynamical properties of specific classes of nonlinear Schrödinger equations. The fractional in time nonlinear Schrödinger equation provides us with a general point of view on the relationship between the effects of absorption and dispersion terms, in optical soliton systems evolving over Lévy trajectories. This subject may be relevant for the approximate and exact controllability of certain nonlinear equations and their solutions. Here, we obtained the nonexistence of global weak solutions to problems (1) and (2), adopting the Pohozaev nonlinear capacitary method. Consequently, we discussed in details some particular choices of absorption and dispersion terms, also with the help of examples.