Cauchy problem for the complex Ginzburg-Landau type Equation with L p -initial data

Daisuke Shimotsuma; Tomomi Yokota; Kentarou Yoshii

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 353-361
  • ISSN: 0862-7959

Abstract

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This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation ∂ u ∂ t - ( λ + i α ) Δ u + ( κ + i β ) | u | q - 1 u - γ u = 0 in ℝ N × ( 0 , ∞ ) with L p -initial data u 0 in the subcritical case ( 1 ≤ q < 1 + 2 p / N ), where u is a complex-valued unknown function, α , β , γ , κ ∈ ℝ , λ > 0 , p > 1 , i = - 1 and N ∈ ℕ . The proof is based on the L p - L q estimates of the linear semigroup { exp ( t ( λ + i α ) Δ ) } and usual fixed-point argument.

How to cite

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Shimotsuma, Daisuke, Yokota, Tomomi, and Yoshii, Kentarou. "Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data." Mathematica Bohemica 139.2 (2014): 353-361. <http://eudml.org/doc/261909>.

@article{Shimotsuma2014,
abstract = {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 ($1\le q< 1+2p/N$), 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 $\lbrace \exp (t(\lambda +\{\rm i\} \alpha )\Delta )\rbrace $ and usual fixed-point argument.},
author = {Shimotsuma, Daisuke, Yokota, Tomomi, Yoshii, Kentarou},
journal = {Mathematica Bohemica},
keywords = {local existence; complex Ginzburg-Landau equation; local existence; complex Ginzburg-Landau equation},
language = {eng},
number = {2},
pages = {353-361},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cauchy problem for the complex Ginzburg-Landau type Equation with $L^\{p\}$-initial data},
url = {http://eudml.org/doc/261909},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Shimotsuma, Daisuke
AU - Yokota, Tomomi
AU - Yoshii, Kentarou
TI - Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 353
EP - 361
AB - 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 ($1\le q< 1+2p/N$), 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 $\lbrace \exp (t(\lambda +{\rm i} \alpha )\Delta )\rbrace $ and usual fixed-point argument.
LA - eng
KW - local existence; complex Ginzburg-Landau equation; local existence; complex Ginzburg-Landau equation
UR - http://eudml.org/doc/261909
ER -

References

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