Nonexistence of global solutions for fractional temporal Schrödinger equations and systems
Date
2017-11-08
Authors
Azman, Ibtehal
Jleli, Mohamed
Kirane, Mokhtar
Samet, Bessem
Journal Title
Journal ISSN
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.