Flat blow-up solutions for the complex Ginzburg Landau equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916443500576768 |
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| author | Duong, Giao Ky Nouaili, Nejla Zaag, Hatem |
| author_facet | Duong, Giao Ky Nouaili, Nejla Zaag, Hatem |
| contents | In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i β) Δu + (1 + i δ) |u|^{p-1}u - αu, \quad \text{where } β, δ, α\in \mathbb{R}. $$ 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 \(β\in \mathbb{R}\), 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. \cite{POPphd98}, when \(β= 0\) and \(δ\) is large, blow-up does not occur for \textit{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 \(δ\in \mathbb{R}\) with different types of blowup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02297 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flat blow-up solutions for the complex Ginzburg Landau equation Duong, Giao Ky Nouaili, Nejla Zaag, Hatem Analysis of PDEs 35K05, 35B40, 35K55, 35K57 In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i β) Δu + (1 + i δ) |u|^{p-1}u - αu, \quad \text{where } β, δ, α\in \mathbb{R}. $$ 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 \(β\in \mathbb{R}\), 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. \cite{POPphd98}, when \(β= 0\) and \(δ\) is large, blow-up does not occur for \textit{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 \(δ\in \mathbb{R}\) with different types of blowup. |
| title | Flat blow-up solutions for the complex Ginzburg Landau equation |
| topic | Analysis of PDEs 35K05, 35B40, 35K55, 35K57 |
| url | https://arxiv.org/abs/2308.02297 |