Blow-Up of Solutions of the Mixed Problem for the Nonlinear Ginzburg–Landau–Schrödinger Evolution Equation

Abstract

Here f(u) = (ω + iγ)|u|; α, β, ω, γ, and p are given real numbers; moreover, α + β 6= 0, ω + γ 6= 0, and p > 0. Equation (1) is used in various fields of applied physics, nonlinear quantum mechanics, and the theory of light wave propagation in nonlinear media (e.g., see [1–3]). For the case in which α = 0, f(u) = iγ|u|u+ iμu, γβ > 0, and pn ≥ 4, the blow-up of solutions of problem (1), (2) was considered in [4] for μ = 0 and in [5] for μ > 0. The blow-up of solutions of the Cauchy problem for Eq. (1) with μ = 0 was studied in [4, 6–10] and other papers.