Nonexistence of global solutions for fractional temporal Schrödinger equations and systems

Date

2017-11-08

Authors

Azman, Ibtehal
Jleli, Mohamed
Kirane, Mokhtar
Samet, Bessem

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Volume Title

Publisher

Texas State University, Department of Mathematics

Abstract

We, first, consider the nonlinear Schrödinger equation iαC0 Dαtu + ∆u = λ|u|p + μα(x) ‧ ∇|u|q, t > 0, x ∈ ℝN, where 0 < α < 1, iα is the principal value of iα, C0 Dαt is the Caputo fractional derivative of order α, λ ∈ ℂ\ {0}, μ ∈ ℂ, p > q > 1, u(t, x) is a complex-valued function, and α : ℝN → ℝN is a given vector function. We provide sufficient conditions for the nonexistence of global weak solution under suitable initial data. Next, we extend our study to the system of nonlinear coupled equations iαC0 Dαtu + ∆u = λ|v|p> + μα(x) ‧ ∇|v|q, t > 0, x ∈ ℝN, iβC0 Dβtv + ∆v = λ|u|k + μb(x) ‧ ∇|u|σ, t > 0, x ∈ ℝN, where 0 < β ≤ α < 1, λ ∈ ℂ\{0}, μ ∈ ℂ, p > q > 1, k > σ > 1, and α, b : ℝN → ℝN are two given vector functions. Our approach is based on the test function method.

Description

Keywords

Fractional temporal Schrödinger equation, Nonexistence, Global weak solution

Citation

Azman, I., Jleli, M., Kirane, M., & Samet, B. (2017). Nonexistence of global solutions for fractional temporal Schrödinger equations and systems. <i>Electronic Journal of Differential Equations, 2017</i>(276), pp. 1-17.

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Attribution 4.0 International

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