Cauchy problem for the complex Ginzburg-Landau type Equation with $L^{p}$-initial data

Shimotsuma, Daisuke ; Yokota, Tomomi ; Yoshii, Kentarou

Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 353-361.

Résumé

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.

Détail

DOI : 10.21136/MB.2014.143860

Classification : 35A01, 35Q55, 35Q56
Keywords: local existence; complex Ginzburg-Landau equation