## Cauchy problem for the complex Ginzburg-Landau type Equation with $$L^{p}$$-initial data.(English)Zbl 1340.35340

Summary: This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $\dfrac {\partial u}{\partial t} -(\lambda +{\text i} \alpha)\Delta u +(\kappa +{\text i} \beta)| u| ^{q-1}u-\gamma u=0$ in $$\mathbb {R}^{N}\times (0,\infty)$$ with $$L^{p}$$-initial data $$u_{0}$$ in the subcritical case ($$1\leqslant q< 1+2p/N$$), where $$u$$ is a complex-valued unknown function, $$\alpha$$, $$\beta$$, $$\gamma$$, $$\kappa \in \mathbb {R}$$, $$\lambda >0$$, $$p>1$$, $${\text i} =\sqrt {-1}$$ and $$N\in \mathbb {N}$$. The proof is based on the $$L^{p}$$-$$L^{q}$$ estimates of the linear semigroup $$\{\exp (t(\lambda +{\text i} \alpha)\Delta)\}$$ and usual fixed-point argument.

### MSC:

 35Q56 Ginzburg-Landau equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence

### Keywords:

local existence; complex Ginzburg-Landau equation
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