On the local and global existence of solution for a general Ginzburg-Landau like equation coupled with a Poisson equation in \(L^p({\mathbb R}^D)\) (Q5933736)

The author examines the existence of a local or global semigroup for a complex Ginzburg-Landau like equation coupled with a Poisson equation. The paper first considers the Cauchy problem in the classical Sobolev spaces \(L^p\). Then weighted Sobolev spaces are considered. Using smoothing properties of the linear part, the author obtains a continuous strong solution in \(W^{1,p}\) with a singularity at \(t= 0\), behaving like \(t^{-1/2}\). Similar results are obtained in weighted Sobolev spaces.