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Keywords:
local existence; complex Ginzburg-Landau equation
local existence; complex Ginzburg-Landau equation
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.
References:
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