On the local and global existence of solution for a general Ginzburg-Landau like equation coupled with a Poisson equation in {$L^p({\Bbb R}^d)$}

Abstract

We investigate the existence of a local or global semi-group for a complex Ginzburg-Landau like equation in $u$ coupled with a Poisson equation in $\phi$ defined on the whole space $\mathbb R^d$. At first, we consider the Cauchy problem in the classical Sobolev spaces $L_{\mathbb C}^{p}(\mathbb R ^{d})$, and later we study it in the weighted Sobolev spaces $L_{\rho\mathbb C }^{p}(\mathbb R ^{d})$, where $p\geq 3/2$, $p\geq d$, $d$ is a positive integer and the weight $\rho$ is increasing. Using the smoothing properties of the linear part, we obtain, for initial data in $L_{\mathbb C }^{p}(\mathbb R ^{d})$, a continuous strong solution in $W_{\mathbb C }^{1,p}(\mathbb R ^{d})$ with a singularity at $t=0$ behaving like $t^{-\frac{1}{2}}$. We obtain analogous results in weighted Sobolev spaces.