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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
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