2023/08/04 by Giao Ky Duong, Duong, Giao Ky, Nejla Nouaili +3
Mathematics · Physics and Astronomy · #35B40 #35K05 #35K55 #35K57 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2308.02297
openalex publication_date 2023/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we consider the complex Ginzburg-Landau equation ∂t u = (1 + i β) Δu + (1 + i δ) |u|p-1u - αu, where β, δ, α∈ ℝ. The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for \(β\) and \(δ\). Specifically, for a fixed \(β∈ ℝ\), the existence of finite-time blow-up solutions for arbitrarily large values of \( |δ| \) is still unknown. According to a conjecture made by Popp et al. \citePOPphd98, when \(β= 0\) and \(δ\) is large, blow-up does not occur for generic initial data. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for \(β= 0\) and any \(δ∈ ℝ\) with different types of blowup.