## Differential and Integral Equations

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

Seifeddine Snoussi

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 13, Number 1-3 (2000), 61-98.

Dates
First available in Project Euclid: 21 December 2012

Snoussi, Seifeddine. On the local and global existence of solution for a general Ginzburg-Landau like equation coupled with a Poisson equation in {$L^p({\Bbb R}^d)$}. Differential Integral Equations 13 (2000), no. 1-3, 61--98. https://projecteuclid.org/euclid.die/1356124290