Daisuke Shimotsuma, Tomomi Yokota, Kentarou Yoshii, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, e-mail: j1112610ed.tus.ac.jp@gmail.com, yokota@rs.kagu.tus.ac.jp, yoshii@ma.kagu.tus.ac.jp
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+ i \alpha)\Delta u +(\kappa+ 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, $\alpha$, $\beta$, $\gamma$, $\kappa\in\mathbb{R}$, $\lambda>0$, $p>1$, $ i =\sqrt{-1}$ and $N\in\mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{\exp(t(\lambda+ i \alpha)\Delta)\}$ and usual fixed-point argument.
Keywords: local existence; complex Ginzburg-Landau equation
Classification (MSC 2010): 35Q56, 35A01
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