| Title:
|
Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data (English) |
| Author:
|
Shimotsuma, Daisuke |
| Author:
|
Yokota, Tomomi |
| Author:
|
Yoshii, Kentarou |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
353-361 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $$ \dfrac {\partial u}{\partial t} -(\lambda +{\rm i} \alpha )\Delta u +(\kappa +{\rm i} \beta )|u|^{q-1}u-\gamma u=0 $$ in $\mathbb {R}^{N}\times (0,\infty )$ with $L^{p}$-initial data $u_{0}$ in the subcritical case ($1\leq q< 1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha $, $\beta $, $\gamma $, $\kappa \in \mathbb {R}$, $\lambda >0$, $p>1$, ${\rm i} =\sqrt {-1}$ and $N\in \mathbb {N}$. The proof is based on the $L^{p}$-$L^{q}$ estimates of the linear semigroup $\{\exp (t(\lambda +{\rm i} \alpha )\Delta )\}$ and usual fixed-point argument. (English) |
| Keyword:
|
local existence |
| Keyword:
|
complex Ginzburg-Landau equation |
| MSC:
|
35A01 |
| MSC:
|
35Q55 |
| MSC:
|
35Q56 |
| idZBL:
|
Zbl 06362264 |
| idMR:
|
MR3238845 |
| DOI:
|
10.21136/MB.2014.143860 |
| . |
| Date available:
|
2014-07-14T08:40:05Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143860 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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