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On the Ginzburg-Landau wave equation. (English) Zbl 0663.35095
Consider the initial value problem of the Ginzburg-Landau wave equation with a general power self-interaction term: $(*)\quad \phi_ t=(1+i\alpha)\Delta \phi +(1+i\beta)\phi -(1+i\gamma)| \phi |^{\mu -1}\phi,\quad x\in {\mathbb{R}}^ n,\quad t>0,$
$\phi (x,0)=\phi_ 0(x),\quad x\in {\mathbb{R}}^ n,$ where $$\phi$$ is a complex scalar function, $$\alpha$$,$$\beta$$,$$\gamma\in {\mathbb{R}}$$, $$\mu\geq 2$$, $$i=\sqrt{-1}$$, and $$n=1,2,3$$. It is shown that for $$\mu \leq 1+4/n$$ and any $$\phi_ 0\in H^ 2({\mathbb{R}}^ n)$$, Eq. (*) has a unique global solution $$\phi \in C([0,\infty);H^ 2({\mathbb{R}}^ 2))\cap C^ 1([0,\infty);L^ 2({\mathbb{R}}^ n)).$$
Reviewer: Y.S.Yang

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35G25 Initial value problems for nonlinear higher-order PDEs
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