Nonexistence of solutions of higher-order nonlinear non-Gauge Schrödinger equation

Ahmed Alsaedi,Bashir Ahmad,Mokhtar Kirane,Abderrazak Nabti

doi:10.5269/bspm.47911

Abstract

A nonexistence result is proved of the space higher-order nonlinear Schrodinger equation where m > 1, n > 1 and p > n. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided.Next, we extend our results to the 2 × 2 – system.

Highlights

There are few results about upper estimates of the life span of solutions, and necessary conditions of local or global existence for nonlinear Schrodinger equations; we mention that in [9], an upper bound of the life span of solutions for the equation i∂tu + ∆u = λ|u|p, in RN × R+, p > 1, (1.4)
We obtain an upper estimate for the life span of solutions of equation (1.1) with initial data of the form u(x, 0) = μf (x)
Necessary conditions for local or global existence Here, we suppose that the data f satisfy the assumption (H3) f1 ∈ L∞(RN ), λ2f1(x) dx ≥ 0, or f2 ∈ L∞(RN ), λ1f2(x) dx ≤ 0

Summary

Whereupon μ

Tμ ≤ Cμ1/ρ; this completes the proof of the Theorem. 4. Tμ ≤ Cμ1/ρ; this completes the proof of the Theorem. 4. Necessary conditions for local or global existence Here, we suppose that the data f satisfy the assumption (H3) f1 ∈ L∞(RN ), λ2f1(x) dx ≥ 0, or f2 ∈ L∞(RN ), λ1f2(x) dx ≤ 0. The necessary conditions for the existence of local or global solutions to problem (1.1)–(1.2) are presented; these conditions depend on the behavior of the initial condition at infinity. If u is a global solution to problem (1.1)–(1.2), there is a positive constant C > 0 such that lim inf. Let u be a global weak solution to problem (1.1)–(1.2). If we take τ = t/R2m, y = x/R, and use inequality (3.4) in the right-hand side of (4.3), and take account of the positivity of the first term in the left-hand side, we get

Using the estimate

We set