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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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